Abstract
We give a Fourier-theoretic analysis of the convergence of semi-discrete Radial Basis Function interpolation on regular grids and of the associated semi-discrete collocation schemes for evolution equations whose generator is a constant-coefficient pseudo-differential operator. We examine convergence in Wiener norm for a general class of basis functions which generalizes a class introduced earlier by M. Buhmann. We also discuss approximate approximation properties, in the sense of V. Maz’ya and G. Schmidt, both for interpolation and for the collocation scheme. Despite the use of the Wiener norm for the error we can allow a broad class of polynomially increasing functions. Our results apply to parabolic equations such as the heat equation, but also to non-local equations such as the Kolmogorov equation of a multi-dimensional Lévy-process, and to hyperbolic equations such as the (half-)wave equation or the transport equation.
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1 Introduction
Radial basis function (RBF) interpolation constructs interpolating functions as linear combinations of translates of a given basis function \(\varphi . \) Given a finite or countable set of interpolation points \(X = \{ x_j \} \subset {\mathbb {R}}^n \) and data \(\{ f_j \} \), one looks for coefficients \(c_k \) such that \(s_X (x) = \sum _k c_k \varphi (x - x_k ) \) satisfies \(s_X (x_j ) = f_j . \) This translates into a system of linear equations for the coefficients which can be uniquely solved if the coefficient matrix \((\varphi (x_j - x_k ) )_{j, k } \) is non-singular. This will for example be the case for any choice of points \(x_j \) if \(\varphi \) is strictly positive definite. More generally, \(\varphi \) can be taken to be conditionally positive definite provided we add a polynomial to s(x) and suitably augment the linear system: see for example Buhmann [5], Fasshauer [14] and Wendland [36] for the definition of conditional positive definiteness and for general introductions to RBF interpolation, and also Chen et al. [11], specifically for applications to linear PDEs. RBF-interpolation can be turned into a scheme for solving boundary value problems for PDEs on a domain by requiring that s(x) solve the PDE exactly in a set of trial points in the interior of the domain, while satisfying the boundary conditions in a second set of trial points on its boundary. This RBF-collocation method, originally proposed by Kansa [18], is found to perform well numerically. Its convergence was analyzed by Schaback [32] in the more general context of kernel-based methods, where one replaces \(\varphi (x - y) \) by a more general kernel K(x, y).
In this paper we analyse the convergence of a semi-discrete variant of this collocation scheme for evolution equations \(\partial _t u (x, t ) = A u (x, t ) \) on \({\mathbb {R}}^n \) with given initial value at \(t = 0 . \) The scheme we examine is a variant of the classical method of lines: we take the coefficients to be time-dependent and ask that \(s_X (x, t ) = \sum _k c_k (t) \varphi (x - x_k ) \) satisfy the evolution equation in each of the interpolation points \(x_j \) for all positive times: \(\partial _t s_X (x_j , t ) = A (s_X (\cdot , t ) ) (x_j ) \), while interpolating the initial value in these points at \(t = 0 . \) This now leads to a system of ordinary differential equations for the coefficients, linear if A is, which can be solved numerically to high accuracy. This method was originally also proposed and tested by Kansa [18] and has been used by many other authors since: see for example [1, 16, 19, 20, 27, 30, 34] for applications to (mostly linear) PDEs and [3, 9, 10, 38] for non-local evolution equations. The papers [29, 33] combine this scheme with domain decomposition in order to speed up computations, by using the method locally on subdomains, and pasting together the local solutions via a partition of unity.
One would like to analyse the convergence of this semi-discrete scheme as the so-called fill-distance \(h_X = \sup _x \inf _j | x - x_j | \) tends to 0, assuming that the system of ODEs for the coefficients can be solved exactly: in practice the numerical solution of such a system will lead to further discretisation errors which will not be considered in this paper. In RBF interpolation one distinguishes between stationary and non-stationary interpolation. Non-stationary interpolation uses the same basis function \(\varphi \) at all scales \(h_X \) while for stationary interpolation one scales the basis function with the separation distance \(q_X := \frac{1 }{2 } \inf _{j \ne k } | x_j - x_k | \), taking X-dependent basis functions of the form \(\varphi _X (x) := \varphi (x/ q_X ) \) for some fixed \(\varphi . \) The advantage of stationary interpolation is that the norm of the inverse of the coefficient matrix \((\varphi _X (x_j - x_k ) )_{j, k } \) can be bounded independently of X if we assume that \(h_X \) is comparable to \(q_X \): cf. [31]. This is especially clear for interpolation on a regular grid \(X = h {\mathbb {Z}}^n \) with separation distance (or grid-size) h, for which \(\varphi _X (x) = \varphi (x / h ) \) and the coefficient matrix becomes independent of h.
Convergence of the non-stationary semi-discrete RBF scheme was investigated in [22, 37] for the heat equation on the sphere, and in [17] for the heat equation on a bounded interval with Dirichlet boundary conditions, where all three papers used basis functions whose associated native space is a Sobolev space of given order. In [29] the convergence of such a scheme in combination with domain decomposition was examined in the case of a convection-diffusion equation. In this paper we examine the convergence of the stationary variant of the scheme. We do this not only for parabolic equations such as the heat equation, but more generally for evolution equations on \({\mathbb {R}}^n \) whose generator A is a constant coefficient pseudo-differential operator. As such, our results will also apply to non-parabolic evolution equations such as the Schrödinger equation and to evolution equations whose generator is a non-local operator, such as the fractional heat equation and more generally the Kolmogorov forward and backward equations of Lévy-processes. The latter have applications in mathematical finance, and their numerical treatment is an important issue.
Instead of concentrating on specific examples of basis functions we will examine convergence of the scheme for an axiomatically defined class of basis functions \(\varphi \) which is a slight generalisation of a class introduced earlier by Buhmann in [4, 5], and which for that reason we will call the Buhmann class. Its main feature is that the restriction of the (distributional) Fourier transform of \(\varphi \) to should be a positive function \(\widehat{\varphi } (\xi ) \) which, together with a certain number of its derivatives, should decay sufficiently rapidly at infinity and which, most importantly, should have a \(|\xi |^{-\kappa } \)-type of singularity at \(\xi = 0 \), with a \(|\xi |^{- \kappa - |\alpha | } \)-type behavior for derivatives of order \(|\alpha | . \) Such \(\varphi \) can have a polynomial growth at infinity (if \(\kappa \ge n \)). Commonly studied examples of radial basis functions, such has the generalized multi-quadrics, or the polyharmonic basis functions, all belong to such a class. We will show that, under suitable conditions on the initial value f (about which more below), the stationary semi-discrete RBF-scheme for the heat equation \(\partial _t u = \Delta u \) on \({\mathbb {R}}^n \) on uniform grids \(h {\mathbb {Z}}^n \) will converge in Wiener-norm to the true solution, with an error whose Wiener norm \(e_{\varphi } (h) \) can be bounded by \(C h^{\kappa - 2 } \), assuming that \(\kappa \ge 2 . \) This discretisation error \(e_{\varphi } (h) \) is in fact of order exactly \(h^{\kappa - 2 } \), and one can explicitly compute \(\lim _{h \rightarrow 0 } h^{- (\kappa - 2 ) } e_{\varphi } (h) =: e_{\varphi } \) as function of the basis function \(\varphi \), assuming that \(\lim _{\xi \rightarrow 0 } |\xi |^{\kappa } \widehat{\varphi } (\xi ) \) exists (if not, one can compute the \(\limsup \) and \(\liminf \)). More generally, for evolution equations \(\partial _t u = a(D) u \) whose generator a(D) is a constant-coefficient pseudo-differential operator of order q whose symbol \(a(\xi ) \) has non-positive real part, and basis functions \(\varphi \) for which \(\kappa \ge q \), the RBF-scheme will be convergent of order \(h^{\kappa - q } . \) Under a mild assumption on the symbol at at infinity one can again compute \(\lim _{h \rightarrow 0 } h^{-(\kappa - q ) } e_{\varphi } (h) =: e_{\varphi } \) which will also depend on the symbol a.
As already stated, we will prove such estimates for the Wiener norm (the \(L^1 \)-norm of the Fourier transform) of the discretisation error, which of course trivially implies the corresponding sup-norm estimates. We will be able to do so despite the fact that neither the intial condition f nor the solutions of the evolution equation or the approximate solutions provided by the RBF-scheme will necessarily have integrable Fourier transforms. As a necessary preliminary, we will first examine convergence in Wiener norm of ordinary stationary RBF interpolation of functions. Our arguments here differ from those normally used for sup-norm convergence, and we will prove a convergence theorem which has a non-empty intersection with classical theorems of Buhmann and Powell [5] for sup-norm convergence, generalizing these in certain aspects.
For the scheme to convergence in the case of the heat equation, the initial value f will have to be a continuous function of at most polynomial growth, of order strictly less than \(\kappa \), and the restriction of its (distributional) Fourier transform to will have to be integrable with respect to the weighted measure \(\min (|\xi |^{\kappa - 2 } , |\xi |^{\kappa } ) d \xi . \) This imposes a certain smoothness but allows polynomial growth at infinity. Examples of such functions are smooth functions f whose derivatives satisfy the bound \(|\partial ^{\alpha } _x f(x) | \le C_{\alpha } (1 + |x | )^{p - |\alpha | } \) for some \(p < \kappa . \) For an evolution equation whose generator is a pseudo-differential operator of order q with smooth symbol, \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \) should be integrable with respect to \(|\xi |^{\kappa } d\xi \), except when the generator of the evolution operator is elliptic of order q, in which case integrability with respect to \(\min (|\xi |^{\kappa - q } , |\xi |^{\kappa } ) \) will suffice. This is consistent with the gain of q derivatives for solutions of elliptic equations. If the symbol is allowed to have a \(|\xi |^q \)-type of singularity at 0, as in the case of the fractional Laplacian, there are further restrictions on the growth of f: its growth at infinity should be of polynomial order strictly less than q and \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \) should be in \(L^1 ( {\mathbb {R}}^n , \max (|\xi |^q, |\xi |^{\kappa } ) ) . \)
Depending on the basis function, \(\overline{e }_{\varphi } := \limsup _{h \rightarrow 0 } h^{-(\kappa - q ) } e_{\varphi } (h) \) can become very small: if we introduce a shape parameter c, and work with basis functions of the form \(\varphi (x) = \phi (x/c ) \), \(\overline{e }_{\varphi } \) is for example found to decrease exponentially to 0 as \(c \rightarrow \infty \) if \(\phi (x) \) is a generalized multi-quadric. This implies that the scheme can have an apparant rate of convergence which is faster than \(h^{\kappa - q } \), depending on the smoothness of the initial condition f: we will show that if \(|\xi |^s \widehat{f } (\xi ) \) is integrable at infinity, then the error can be bounded by \(2\overline{e }_{\varphi } h^{\kappa - q } + C h^s \), for \(h \le h_0 \) sufficiently small, where in situations where \(\overline{e }_{\varphi } \) decreases exponentially in the shape parameter c, the constant C only increases polynomially. This implies that for h’s which are "small but not too small", the convergence would appear to be of order \(h^s \) instead of \(h^{\kappa - q } . \) Note that this applies in particular when \(\kappa =q \), when there is no true convergence. Similar remarks already apply to the convergence of stationary RBF interpolants of a given function. These are examples of approximate approximation, a notion which was introduced by Maz’ya [24] in the context of stationary quasi-interpolation using Gaussian kernels: see also Maz’ya and Schmidt [25, 26]. Our results generalize the work of Maz’ya and Smith to exact interpolation for a wide class of basis functions. Approximate approximation phenomena were encountered empirically in [3, 9], and the wish to find a theoretical explanation for the numerical results of these papers was one of the motivations for the present paper.
For homogeneous basis functions, \(\varphi (x) = |x |^p \) on \({\mathbb {R}}^n \) (\(p \notin 2 {\mathbb {N}} \)), scaling by a shape parameter c will have no effect, nor is there then a difference between stationary and non-stationary interpolation. There is however a dimensional effect. Since the Fourier transform of \(\varphi \) is a constant times \( |\xi |^{ - p - n } \) (the same is true for \(\varphi (x) = |x |^p \log |x | \) when \(p \in 2 {\mathbb {N}} \)), these functions are Buhman class with a \(\kappa = p + n \) which grows with the dimension n. The semi-discrete RBF scheme will therefore converge more rapidly in higher dimensions. Such an increase of convergence rates with dimension was observed empirically in Chen et al. [11] for the Dirichlet problem for the Poisson equation on a cube with \(|x |^5 \) as basis function. The authors state that, while known for stationary interpolation [5], this phenomenon has no theoretical basis yet when solving partial differential equations. Our results provide one, at least in the context of constant coefficient evolution equations on \({\mathbb {R}}^n . \)
A rate of convergence which increases with the dimension also occurs for non-homogeneous basis functions such as the generalized multi-quadrics \((1 + |x |^2 )^{\nu } \), \(\nu \notin {\mathbb {N}} \), for which \(\kappa = - 2\nu - n \) (see for example [36] for its Fourier transform), and implies that for high-dimensional problems one can use coarser grids to attain a given accuracy. In this respect, the RBF scheme behaves better than Finite Difference and Finite Element methods, though it should be noted that for this dimensional effect to hold, the degree of smoothness of the initial condition f also has to increase: for example, for the heat equation, \(\widehat{f } (\xi ) |\xi |^{\kappa - 2 } \) has to be integrable at infinity.
We finally note another convergence-enhancing effect: for pseudo-differential evolution equation with non-elliptic symbols, \(e_{\varphi } = \lim _{h \rightarrow 0 } h^{-(\kappa - q ) } e_{\varphi } (h) \), which depends on the symbol a, can be 0 if a has a certain symmetry, for example, if it is an odd function (a then necessarily is imaginary-valued, since its real part has to be non-negative). A concrete example is provided by the transport equation.
We expect our results for the convergence rates to generalize to non-uniformly distributed interpolation centers whose fill-distance is comparable to the separation distance, but this will require other methods of proof, since we make heavy use of the translation invariance of the interpolation sets \(h {\mathbb {Z}}^n \) (as well as that of the generator of the evolution equation). We believe that our results for regular grids have an interest of their own, since we get rather precise results, not only for the convergence rates, in terms of powers of h, but also for the multiplicative constants by which these powers of h are multiplied, leading naturally to approximate approximation estimates. Numerical tests of RBF collocation schemes often simply use uniform grids. Adaptive methods have been proposed for variable coefficient and non-linear PDEs, taking advantage of the flexibility of RBF interpolation with regard to the choice of interpolation points, but for constant coefficient pseudo-differential equations on low-dimensional euclidean spaces, regular grids seem a reasonable choice.
Even on regular grids, the RBF-scheme has advantages over traditional methods such as Finite Differences. First of all, it does not require the operators to be discretized: it suffices to compute their action on the basis function. Discretisation of operators may lead to loss of accuracy when one applies Finite Differences to singular integral operators, such as the generators of certain Lévy processes: see [3] for literature references and further discussion. Furthermore, there are the approximate approximation and dimensional effects which can enhance accuracy, in the case of the former for small but not too small h: see also the discussion in Example 6.11 below.
The paper suggests a number of directions for future research, first of all the extension of its results to non-uniformly distributed interpolation centers, to variable coefficient PDEs and to mixed initial-boundary value problems. We mention that for evolution equations on \({\mathbb {R}}^n \) with variable-coefficient generators, implementation of the RBF-scheme on \((h {\mathbb {Z}} )^n \) leads to a pseudo-differential equation on the torus \({\mathbb {R}}^n / 2 \pi h {\mathbb {Z}}^n \) which, in the constant coefficient case considered here, is a simple multiplication operator. This will be examined elsewhere. For boundary-value problems, the standard RBF-approximations will be less accurate near the boundary, and may have to be modified: cf. [15, 17] uses kernel-based interpolation with a kernel which automatically satisfy the boundary conditions.
The methods of this paper can also be applied to examine convergence in Sobolev norm and convergence of the non-stationary version of the scheme on regular grids. With non-stationary interpolation the Fourier transform of the basis function no longer needs to be singular at the origin for the scheme to convergence. One furthermore expects to see spectral convergence (exponentially small discretisation errors) if the Fourier transform of the basis function decays exponentially at infinity, at least for suitably chosen initial values: cf. Madych and Nelson [23] for the spectral convergence of non-stationary RBF-interpolation with multiquadrics for functions in the corresponding native spaces. These questions will also be treated elsewhere.
The paper is organized as follows: in Sect. 2 we define the Buhmann class and discus the associated Lagrange or cardinal function for interpolation an regular grids. Section 3 examines convergence of stationary RBF interpolation in Wiener norm with Buhmann-class functions and Sect. 4 examines approximate approximation in this setting. These sections are a necessary preliminary for the convergence analysis of the semi-discrete collocation scheme in the next two sections. In Sect. 5 we first give a detailed treatment of the scheme for the classical heat equation, since we believe that this is the example most readers would want to see first. We discuss both convergence and approximate approximation. The final section extends the results of Sect. 5 to evolution equations with general translation-invariant pseudo-differential generators. Two appendices contain some more technical material: the proof of the existence and basic properties of the Lagrange functions in appendix A, and an important lemma identifying the Fourier transforms of the RBF-interpolants, and of the approximate solutions as computed by the RBF-scheme, in Appendix B, for cases where the initial condition can have polynomial growth.
We finally would like to thank the referees for their thoughtful comments and for the additional references they provided.
1.1 Notations
C denotes the usual variable constant, whose exact numerical value is allowed to change from one occurrence to another. We use the following convention for the Fourier transform \(\widehat{f } = \mathcal {F }(f) \) of an integrable function f on \({\mathbb {R}}^n \):
\((x, \xi ) \) being the Euclidean inner product on \({\mathbb {R}}^n . \) We will routinely use the extension of the Fourier transform \(\mathcal {F } \) to the space of tempered distributions \(\mathcal {S }' ({\mathbb {R}}^n ) \), where \(\mathcal {S }({\mathbb {R}}^n ) \) is the Schwarz-space of rapidly decreasing functions.
For \(s \in {\mathbb {R}} \), let \(L^1 _s ({\mathbb {R}}^n ) \) be the space of measurable functions f on \({\mathbb {R}}^n \) for which
The homogeneity of the weight \(|\xi |^s \) will be important, since it allows singular behavior of f at 0. For \(r \le s \) we also introduce the "mixed" spaces
where \(a \wedge b := \min (a, b ) . \) This allows the same singularity in 0 while imposing lesser decay at infinity. Note that the scale of spaces is decreasing in r, and that \(L ^1 _{r \wedge s } ({\mathbb {R}}^n ) \subseteq L^1 _{r, s } ({\mathbb {R}}^n ) \), with equality if \(r = s . \)
We finally introduce the weighted sup-norm spaces \(L^{\infty } _s ({\mathbb {R}}^n ) \) as the set of measurable functions for which
where the sup is the essential supremum, as usual. If \(s < 0 \), an element f of \(L^{\infty } _s ({\mathbb {R}}^n ) \) is of polynomial growth of order at most |s| : \(|f(x) | \le C (1 + |x|)^{|s | } \) on \({\mathbb {R}}^n \), with \(C = || f ||_{\infty , s } . \)
Derivatives of functions \(f = f(x) \) on \({\mathbb {R}}^n \) will be denoted by \(\partial _x ^{\alpha } f (x) \) or by \(f^{(\alpha ) } (x) \), \(\alpha \in {\mathbb {N}}^n \) a multi-index. If \(K \in {\mathbb {N}} \) and \(\lambda \in (0, 1 ] \), then \(C_b ^{K, \lambda } ({\mathbb {R}}^n ) \) will denote the Hölder space of K-times differentiable functions on \({\mathbb {R}}^n \) with bounded derivatives of all orders, such that the derivatives of order K satisfy a uniform Hölder condition on \({\mathbb {R}}^n \) with exponent \(\lambda \), provided with the norm
where \(|| g ||_{0 ; \lambda } := \sup _{\xi \ne \eta } | g (\xi ) - g (\eta ) | / |\xi - \eta |^{\lambda } . \)
Finally, \(\lfloor x \rfloor \) and \(\lceil x \rceil \) denote the usual floor and ceiling functions, defined as the greatest, respectively smallest integer which is less than, respectively greater than a real number x; note that \(\lceil x \rceil = \lfloor x \rfloor + 1 \) if \(x \notin {\mathbb {N}} \), while \(\lceil x \rceil = \lfloor x \rfloor = x \) otherwise.
2 A Class of Basis Functions for Interpolation on Regular Grids
2.1 The Buhmann Class
We introduce a flexible class of basis functions which is well-suited for stationary interpolation on regular grids. Since this class of functions is a generalisation of the one introduced by Buhmann [4] (called admissible in there) we will call it the Buhmann class. From the onset, we will allow non-radial basis functions, radiality not being essential for most of the theory.
Definition 2.1
For \(\kappa \ge 0 \) and \(N > n \) we define the Buhmann class \(\mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) as the set of functions \(\varphi \in C({\mathbb {R}}^n ) \) such that
(i) \(\varphi \) is of polynomial growth of order strictly less than \(\kappa \), in the sense that \(\varphi \in L^{\infty } _{- \kappa + \varepsilon } ({\mathbb {R}}^n ) \) for some \(\varepsilon > 0 . \)
(ii) Regularity and strict positivity The restriction to \({\mathbb {R}}^n \setminus 0 \) of the Fourier transform \(\widehat{\varphi } := \mathcal {F } (\varphi ) \) (in the sense of tempered distributions) can be identified with a function in \(C^{n + \lfloor \kappa \rfloor + 1 } ({\mathbb {R}}^n \setminus 0 ) \), which we will continue to denote by \(\widehat{\varphi } \), and which is pointwise strictly positive: \(\widehat{\varphi } (\eta ) > 0 \) for all \(\eta \in {\mathbb {R}}^n \setminus 0 . \)
(iii) Elliptic singularity at 0 There exist positive constants c, C such that for all \(|\alpha | \le n + \lfloor \kappa \rfloor + 1 \),
while also
(iv) Decay at infinity There exist positive constants \(C_{\alpha } \), \(|\alpha | \le n + \lfloor \kappa \rfloor + 1 \), such that
We will call the parameter \(\kappa \) the order of the singularity (at 0) and N the decay rate (at infinity) of the Buhmann class \(\mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \), with the understanding that both of these refer to the Fourier transform of an element of the class. We use the term "elliptic" for condition (iii) because of the resemblance of (4) and (5) with the ellipticity condition on symbols of pseudo-differential operators (where the singularity would be at infinity). The significance of \(n + \lfloor \kappa \rfloor + 1 \) is that this is the smallest integer which is strictly greater than \(n + \kappa . \) (Note that if \(\kappa \notin {\mathbb {N}} \), then \(n + \lfloor \kappa \rfloor + 1 = n + \lceil \kappa \rceil . \)) Conditions (ii) and (iii) for derivatives up till this order will imply polynomial decay of order \(n + \kappa \) of the associated Lagrange interpolation function which we will define below. Requiring higher order differentiability would not improve this rate of decay: \(n + \kappa \) is best possible, under condition (iii).
In most of the results of this paper, strict positivity of \(\widehat{\varphi } \) on \({\mathbb {R}}^n \setminus 0 \) could have been replaced by the weaker condition that the "periodisation" \(\sum _k \widehat{\varphi } (\eta + 2 \pi k ) \) of \(\widehat{\varphi } \) be pointwise strictly positive on all of \({\mathbb {R}}^n \), as in [4]; note that by (6) with \(\alpha = 0 \), this series converges absolutely on \({\mathbb {R}}^n \setminus {\mathbb {Z}}^n \) , given that \(N > n \), while it can be set equal to \(\infty \) on \({\mathbb {Z}}^n \), in view of (5). Since for most of the radial basis functions used in practice, \(\widehat{\varphi } (\eta ) \) itself is already strictly positive, we have opted to impose the stronger condition, which also simplifies the proofs.
Remarks 2.2
(i) Buhmann [4] studied stationary RBF interpolation on regular grids for a slightly more restricted class of radial basis functions. The main difference between his original class and the one of our Definition 2.1 (besides, as already mentioned, Buhmann requiring strict positivity of the periodisation of \(\widehat{\varphi } \) instead of of \(\widehat{\varphi } \) itself) lies in condition (iii), where Buhmann asks that for small \(|\eta | \), \(\widehat{\varphi }(\eta ) \) be asymptotically equivalent to a positive multiple of \(|\eta |^{- \kappa } \) modulo an relative error which has to be sufficiently small: \(\widehat{\varphi } (\eta ) = A |\eta |^{- \kappa } (1 + h(\eta ) ) \) with \(A > 0 \) and \(|\partial ^{\alpha } _{\eta } h(\eta ) | = O(|\eta |^{\varepsilon - |\alpha | } ) \) as \(\eta \rightarrow 0 \) for \(|\alpha | \le n + \lfloor \kappa \rfloor + 1 \), with an \(\varepsilon > \lceil \kappa \rceil - \kappa . \) Under these conditions Buhmann proved the existence of a unique Lagrange function for interpolation on \({\mathbb {Z}}^n \), constructed as an infinite linear combination of translates of \(\varphi \), which moreover decays as \(|x |^{- \kappa - n } \) at infinity. This fundamental result remains true for \(\varphi \)’s in \(\mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \): see Theorem 2.3 below and and its proof in Appendix A. The condition that \(\varepsilon > \lceil \kappa \rceil - \kappa \) is in our treatment made unnecessary by Lemma A.2.
(ii) All conditions in Definition 2.1 except the first are on the Fourier transform of \(\varphi . \) One can show (cf. Appendix A) that if the Fourier transform of a polynomially increasing function \(\varphi \) satsifies (ii), (iii) and (iv), then there exists a function \(\widetilde{\varphi } (x) \) which grows at most as \(\max (|x |^{\kappa - n } \log |x| , 1 ) \) at infinity (and, slightly better, as \(\max (|x|^{\kappa - n } , 1 ) \) if \(\kappa \notin {\mathbb {N}} \)) and a polynomial P(x) such that
The function \(\widetilde{\varphi } \) is unique modulo polynomials of degree \(\lfloor \kappa \rfloor - n . \) If we moreover require \(\varphi \) to have polynomial growth of order strictly less than \(\kappa \), as in Definition 2.1, then P(x) will be a polynomial of degree of at most \(\lceil \kappa \rceil - 1 \) (which is \(\lfloor \kappa \rfloor \) if \(\kappa \notin {\mathbb {N}} \), and \(\kappa - 1 \) if \(\kappa \in {\mathbb {N}} \)). Note that the Fourier transform of a polynomial is a linear combination of derivatives of the delta-distribution in 0, and therefore equals 0 on \({\mathbb {R}}^n \setminus 0 . \)
(iii) The condition that \(N > n \) will suffice for convergence of stationary RBF interpolation on regular grids \(h {\mathbb {Z}}^n \) as \(h \rightarrow 0 \), but will have to be strengthened to \(n > N + q \) for convergence of the semi-discrete RBF schemes studied here for solving evolution equations whose spatial part is a partial- or pseudo-differential operator of order q.
The usual examples of radial basis functions, such as the generalised multi-quadrics, cubic and higher order splines, thin plate splines, inverse multi-quadrics and Gaussians, are all Buhmann class.
One can show that if \(\varphi \in L^{\infty } _{- p } ({\mathbb {R}}^n ) \), \(p \in {\mathbb {N}} \), satisfies conditions (ii)–(iv) of Definition 2.1, then \(\varphi \) is conditionally positive definite of order \(\mu \), where \(\mu \) is the smallest integer such that \(2 \mu > \max ( \lfloor \kappa \rfloor - n , p , 0 ) \): for this it would in fact be sufficient that \(\widehat{\varphi } |_{{\mathbb {R}}^n \setminus 0 } \) is locally integrable, satisfies (4) with \(\alpha = 0 \) and is integrable on \(\{ |\eta | \ge 1 \} . \) One can therefore, by standard RBF theory, interpolate an arbitrary function on a finite set X of points by a linear combination of translates of \(\varphi \) plus a polynomial of degree \(\mu - 1 \), provided the set X is unisolvent for this class of polynomials: see for example [5]. This involves solving a linear system of equations. The next theorem establishes the existence and main properties of a Lagrange function in terms of which the solution of the interpolation problem on \({\mathbb {Z}}^n \) can be simply expressed.
Theorem 2.3
Suppose that \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with singularity of order \(\kappa \ge 0 \) and decay rate \(N > n . \) Then there exist coefficients \(c_k \), \(k \in {\mathbb {Z}}^n \), such that the series
converges absolutely and uniformly on compacta and defines a Lagrange function for interpolation on \({\mathbb {Z}}^n : \)
The function \(L_1 \) satisfies the bound
and its Fourier transform is given by
Moreover, at the points of \(2 \pi {\mathbb {Z}}^n \), \(\widehat{L }_1 \) satisfies the Fix–Strang estimates
See Appendix A for the proof. Observe that
Remarks 2.4
(i) We will write \(L_1 (\varphi ) \) if we want to stress the dependence on the basis function \(\varphi \), otherwise we will simply write \(L_1 . \) The subindex 1 in \(L_1 \) is a notational reminder that \(L_1 \) is a Lagrange function for interpolation on the standard grid \({\mathbb {Z}}^n \) of width 1. For stationary RBF interpolation on the scaled grids \(h {\mathbb {Z}}^n \) one uses the scaled basis functions \(\varphi (x/h ) \), whose associated Lagrange functions then simply are \(L_h (x) := L_1 (x/h ) . \) If \(f \in L^{\infty } _{- p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \), then \(s_h [f ] (x) := \sum _j f(hj ) L_1 (h^{-1 } x - j ) \) is an infinite linear combination of translates of \(\varphi \) which will interpolate f on \(h {\mathbb {Z}}^n \), where the series converges absolutely and uniformly on compacta, in view of the growth restriction on f.
(ii) One important point of the theorem is that the basis function \(\varphi \) need not decay at infinity, but is allowed to grow polynomially. A high order of growth will in fact lead to a high order convergence of the stationary RBF interpolants \(s_h [f ] \) to f as \(h \rightarrow 0 \), since this will translate into a strong singularity in 0 of the Fourier transform of \(\widehat{\varphi } \) in the form of a large \(\kappa \), which implies that \(\widehat{L }_1 \) will satisfy the Fix - Strang conditions to a high order. The latter then implies a convergence rate of \(O(h^{\kappa } ) \) in sup-norm, as shown by Buhmann [4] (under suitable conditions on f); see also [5], Chapter 4. We will prove such convergence theorems for the Wiener norm instead of the sup-norm, using a different method than [4, 5]: cf. Theorems 3.2 and 3.4 below. Note that, contrary to \(\varphi \), the Lagrange function \(L_1 \) will decay at infinity, as shown by (8), and this the more rapidly the higher \(\kappa \) is. In particular, \(L_1 \) is integrable if \(\kappa > 0 \) and its Fourier transform then exists in the classical sense, as an absolutely convergent integral. It is possible for \(L_1 (x) \) to have faster decay: Buhmann [4] shows that if \(\widehat{\varphi } (\eta ) \sim |\eta |^{- \kappa } \) as \(\eta \rightarrow 0 \) with \(\kappa \in 2 {\mathbb {N}} \), then
while there are examples of \(\varphi \) for which \(L_1 (x) \) decays exponentially: see [5] for details and references.
(iii) The Proof of Theorem 2.3 shows that the coefficients \(c_{-k } \) are precisely the Fourier coefficients of \((\sum _k \widehat{\varphi } (\eta + 2 \pi k )^{-1 } . \) They satisfy bounds analogous to the ones satisfied by \(L_1 \), \(|c_k| = O(|k |^{- n - \kappa } ) \), which guarantee that the defining series for \(L_1 (x) \) converges absolutely and uniformly on compacta, including when \(\kappa = 0 \), in view of condition (i) of Definition 2.1.
We state the following lemma for future reference.
Lemma 2.5
There exist constants \(C_{\alpha } \), \(|\alpha | \le n + \lfloor \kappa \rfloor + 1 \), such that for all \(k \in {\mathbb {Z}}^n \),
In particular, if \(\kappa > 0 \) then \(\widehat{L }_1 \) belongs to the Hölder space \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) \), with \(\lambda = \kappa - (\lceil \kappa \rceil - 1 ) . \)
Note that \(\lceil \kappa \rceil - 1 = \lfloor \kappa \rfloor \) if \(\kappa \) is non-integer, but that it is equal to \(\kappa - 1 \) if \(\kappa \) is a positive integer, so that \(\lambda = 1 \) then.
Proof
This is elementary: if we let \(\widehat{\varphi }_\mathrm{per } (\eta ) := \sum _k \widehat{\varphi } (\eta + 2 \pi k ) \), then (12) follows by induction on \(\alpha \), by applying Leibnitz’s rule to \(\widehat{L }_1 \widehat{\varphi }_\mathrm{per } = \widehat{\varphi } \) if \(k \ne 0 \), and to \( \widehat{\varphi }_\mathrm{per } \left( \widehat{L }_1 - 1 \right) = \widehat{\varphi } - \widehat{\varphi }_\mathrm{per } \) if \(k = 0 \), observing that \(\partial _{\eta } ^{\alpha } \widehat{\varphi }_\mathrm{per } (\eta + 2 \pi k ) = \partial _{\eta } ^{\alpha } \widehat{\varphi }_\mathrm{per } (\eta ) = O(|\eta |^{- \kappa - |\alpha | } ) . \) \(\square \)
Remark 2.6
The fact that \(\widehat{L }_1 \) in general only belongs to a certain Hölder space will be important for extending the Wiener-norm convergence of the interpolation and collocation scheme to functions which are themselves not in Wiener space. We briefly pause to examine the differentiability of \(\widehat{L }_1 \) if \(\kappa \in {\mathbb {N}} . \) If we let \(g(\eta ) := \sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k ) \) and \(\psi (\eta ) := |\eta |^{\kappa } \widehat{\varphi } (\eta ) \), then
shows that \(\widehat{L }_1 \) cannot be \(C^{\kappa } \) in 0 unless \(\kappa \in {\mathbb {N}} \) is even, even if \(\psi \) would be (note that \(\psi (0) \ne 0 \) since \(\varphi \) is Buhmann class). If \(\kappa \in 2 {\mathbb {N}} \), then \(\widehat{L }_1 \) will be as smooth as \(\psi \) and \(\widehat{\varphi } \) are.
We finally note that to construct numerical PDE schemes using RBF interpolation one will obviously need sufficient differentiability of \(L_1 . \) The Proof of Theorem 2.3 given in appendix A also yields existence and decay of derivatives of \(L_1 \), provided N is chosen sufficiently large:
Theorem 2.7
Suppose that \(k \in {\mathbb {N}} \) and let \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with decay rate \(N > n + k . \) Then \(L_1 \in C^k ({\mathbb {R}}^n ) . \) Moreover, \(|\partial _x ^{\alpha } L_1 (x) | = O(|x |^{- \kappa - n } ) \) as \(|x| \rightarrow \infty \), for all \(|\alpha | \le k . \)
See also Theorem 6.1 below.
3 Convergence of RBF-Interpolants
In this section we re-examine the convergence of stationary RBF interpolation on a regular grid \(h {\mathbb {Z}}^n \), as a preliminary to our analysis of the semi-discrete RBF scheme. Our approach differs from that of [4, 5]: instead of basing the analysis on exact reproduction of polynomials up to a certain degree (in combination with Taylor’s formula) we estimate the Wiener norm \(|| s_h [f ] - f ||_A \) of the error, where \(s_h [f ] \) is the RBF-interpolant of f and where we recall that \(|| f ||_A = || \widehat{f } ||_1 . \) This of course trivially implies estimates in the uniform norm. Computation of the Fourier transform of \(s_h [f] \), together with the properties of the Lagrange function listed in Theorem 2.3, and in particular the Fix–Strang estimates (10), will quickly lead to what will turn out to be a sharp estimate for the interpolation error when \(\widehat{f } \) is integrable. This will be then extended to a class of polynomially growing functions f whose Fourier transform can be singular in 0.
We fix a basis function \(\varphi = \varphi _1 \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) whose Fourier transform has a singularity of order \(\kappa > 0 \) at 0 and a decay rate \(N > n \) at infinity. We interpolate functions on \(h {\mathbb {Z}}^n \) using a basis function which scales with the grid-size: \(\varphi _h (x) := \varphi (x/h ) . \) The associated Lagrange function scales similarly, and the RBF interpolant \(s_h [f] \) of a given function \(f : {\mathbb {R}}^n \rightarrow {\mathbb {C}} \) can be written as
where \(L_1 \) is the Lagrange function of Theorem 2.3. Here, and below, sums over j, k, \(\ell \), etc. are understood to be over \({\mathbb {Z}}^n . \) The decay (8) of \(L_1 (x) \) implies that the series (13) converges absolutely if f is of polynomial growth of order strictly less than \(\kappa \), that is, if \(f \in L^{\infty } _{- \kappa - \varepsilon } ({\mathbb {R}}^n ) \) for some \(\varepsilon > 0 . \) (If \(\kappa = 0 \) this condition would exclude functions f with integrable Fourier transform, but one can then re-sum the series (13): see Remark 3.9).
3.1 Convergence in Wiener Norm
We begin by computing the Fourier transform of \(s_h [f ] \) for Schwarz-class functions f. For sufficiently rapidly decaying functions g, let us define the function \(\Sigma _h (g) \)
The map \(\Sigma _h : g \rightarrow \Sigma _h (g) \) will play an important rôle in what follows. We note that \(\Sigma _h \) is a contraction with respect to the \(L^1 \)-norm: indeed, by the positivity of \(\widehat{L }_1 \) and monotone convergence,
in view of (11); \(\Sigma _h \) therefore extends to a contraction on \(L^1 ({\mathbb {R}} ) . \) We also note that if \(g \in L^1 ({\mathbb {R}}^n ) \), then the defining series for \(\Sigma _h (g) \) converges absolutely a.e., since
Lemma 3.1
If \(f \in \mathcal {S }({\mathbb {R}}^n ) \) then \(s_h [f ] \in L^1 ({\mathbb {R}}^n ) \) and
Proof
Since \(\kappa > 0 \), \(L_1 \) is integrable by Theorem 2.3 and therefore \(|| s_h [f ] ||_1 \le \left( h^n \sum _j |f(hj ) | \right) || L_1 ||_1 . \) Applying Fubini’s theorem to the function \((j, x ) \rightarrow f(hj ) L_1 (h^{-1 } x - j ) e^{- i (x, \xi ) } \) on \({\mathbb {Z}}^n \times {\mathbb {R}}^n \) one finds
where for the second line we used the Poisson summation formula: \(\sum _j g(j) = \sum _k \widehat{g } (2 \pi k ) \), with \(g(x) := f(hx ) e^{- i h (x, \xi ) } . \) \(\square \)
We can then state a first convergence theorem:
Theorem 3.2
Let \(\kappa > 0 . \) Then there exists a constant \(C = C_{\varphi } > 0 \) such that for all tempered functions f such that \(\widehat{f } \in L^1 ({\mathbb {R}} ) \cap L^1 _{\kappa } ({\mathbb {R}}^n ) \) and for all positive \(h > 0 \),
Proof
We first verify that Lemma 3.1 still holds true if \(\widehat{f } \in L^1 ({\mathbb {R}}^n ) \): f is a bounded continuous function, and the series (13) converges absolutely and uniformly, since \( | s_h [f ](x) | \le || f ||_{\infty } \sum _j | L_1 (h^{-1 } x - j ) | \); the series on the right hand side is h-periodic, and its sup on \(\{ |x | \le h / 2 \} \) can be estimated by a constant times \(\sum _j | L_1 (j) | \) which converges by Theorem 2.3, since \(\kappa > 0 . \) In particular, \(s_h [f] \) is bounded and its Fourier transform exists as a tempered distribution. An easy approximation argument using the density of \(\mathcal {S }({\mathbb {R}}^n ) \) in \(L^1({\mathbb {R}}^n ) \) and the contractivity of \(\Sigma _h \) then shows that \(\widehat{s_h [f] } = \Sigma _h (\widehat{f } ) . \)
It follows that \(\widehat{f } - \widehat{s_h [f ] } = (1 - \widehat{L }_1 (h \xi ) ) \widehat{f } (\xi ) - \sum _{k \ne 0 } \widehat{f }(\xi + 2 \pi h^{-1 } k ) \widehat{L }_1 (h \xi ) \), and since \(0 \le \widehat{L }_1 \le 1 \), montone convergence implies that
where we once more used the identity (11). The Fix–Strang estimate (10) in 0 then implies (17) with \(C = 2 \sup _{\eta \ne 0 } (1 - \widehat{L }_1 (\eta ) ) / |\eta |^{\kappa } \) (a number which, in principle at least, is explicitly computable for a given \(\varphi \)). \(\square \)
Remark 3.3
The theorem generalizes to the case when \(\widehat{f } = \nu \) is a finite Borel measure for which \(|\xi |^{\kappa } \in L^1 ({\mathbb {R}}^n , d |\nu | ) \): in that case,
To show this, one first defines \(\Sigma _h (\nu ) \) by duality: if \(\psi \in \mathcal {S } ({\mathbb {R}}^n ) \), then
where \(\Sigma _h ' (\psi ) := \sum _k \psi (\xi + 2 \pi h^{-1 } k ) \, \widehat{L }_1 (h \xi + 2 \pi k ) \in C_b ({\mathbb {R}}^n ) \), and one checks that \(\widehat{s_h [f ] } = \Sigma _h (\nu ) \) as tempered distributions. Since \(|| \Sigma _h ' (\psi ) ||_{\infty } \le ||\psi ||_{\infty } \), \(\Sigma _h (\nu ) \) is a finite Borel measure. Using (11) again, one estimates
where we can take \(\psi \in C_b ({\mathbb {R}}^n ) . \) It follows that the variation norm of \(\Sigma _h (\nu ) - \nu \) is bounded by \(C h^{\kappa } \), which implies (19).
The conditions on \(\widehat{f } \) at infinity imply a certain smoothness: f will have continuous bounded derivatives of order \(\lfloor \kappa \rfloor . \) On the other hand, the right hand side of (17) can still be finite if \(\widehat{f } \) has a non-integrable singularity at 0. Allowing such singularities means allowing f’s which grow at a certain polynomial rate. We prove the following extension of Theorem 3.2:
Theorem 3.4
Suppose that \(f \in L^{\infty } _{-p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that
Then \(\widehat{s _h [f ] } - \widehat{f } \) is in \(L^1 ({\mathbb {R}}^n ) \), and
for all \(h > 0 . \)
Equation (20) means that the restriction to \({\mathbb {R}}^n \setminus 0 \) of the tempered distribution \(\widehat{f } \) can be identified with a locally integrable function which, when interpreted as an a.e. defined function on \({\mathbb {R}}^n \), is integrable with respect to the weight \(|\xi |^{\kappa } . \)
Proof
Elementary estimates show that \(|| \, s_h [f ] \, ||_{\infty , -p } \le C || \, f \, ||_{\infty , -p } . \) In particular, \(s_h [f] \) is a tempered function. The next lemma, whose somewhat technical proof is given in Appendix B, identifies its Fourier transform.
Lemma 3.5
Suppose that \(|f(x) | \le C (1 + |x | )^p \) for some \(p < \kappa \) and that \(\widehat{f } |_{{\mathbb {R}} \setminus 0 } \in L^1 ({\mathbb {R}}^n , \min ( |\xi |^{\kappa } , 1 ) d\xi ) . \) Then the tempered distribution \(\widehat{s_h [f ] } - \widehat{f } \) can be identified with the function
which is in \(L^1 ({\mathbb {R}}^n ) . \)
The \(L^1 \)-norm of (22) can, as before, be bounded by the \(L^1 \)-norm of \(2 ( \widehat{L }_1 (h \xi ) - 1 ) |\widehat{f } (\xi ) | \) and the theorem follows from the Fix-Strang estimate in \(\xi = 0 . \) \(\Box \)
The proof of the lemma involves extending \(\widehat{f } \) to a continuous linear functional on the Hölder spaces \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) \) with \(\lambda = \kappa - (\lceil \kappa \rceil - 1 ) \) (so that \(\lambda = \kappa - \lfloor \kappa \rfloor \) if \(\kappa \notin {\mathbb {N}} \), and \(\lambda = 1 \) otherwise), and using this to define \(\Sigma _h (\widehat{f } ) \) as a tempered distribution. Note that the individual terms of the series (22) are all integrable, by the Fix–Strang estimates (10).
Example 3.6
It is well-known that if \(f \in C^{\infty }({\mathbb {R}}^n ) \) is such that for some \(p > -n \) and all multi-indices \(\alpha \),
then \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in C({\mathbb {R}}^n \setminus 0 ) \) with
for all \(N \in {\mathbb {N}} \): see for example [35], proposition 1 of Chapter VI, Sect. 4. It follows from the estimate with \(N = 0 \) that \(|\xi |^{\kappa } \widehat{f } (\xi ) \) is integrable at 0 if \(p < \kappa . \) Theorem 3.4 therefore applies to such f. Inspection of the proof shows that for (23) to hold with \(N = 0 \), we only need to require (23) for derivatives up to order \(\lfloor p \rfloor + n + 1 \) and that in that case \(\widehat{f } (\xi ) = O(|\xi |^{ - \lfloor p \rfloor - n - 1 } ) \) at infinity, so that then \(\widehat{f } |_{{\mathbb {R}} \setminus 0 } \in L^1 _{\kappa } ({\mathbb {R}}^n ) \) if \(p< \kappa < \lfloor p \rfloor + 1 . \) For larger \(\kappa \) we need to impose (23) for all \(|\alpha | \le n + \lfloor \kappa \rfloor + 1 \), to ensure the integrability of \(|\xi |^{\kappa } \widehat{f } (\xi ) \) at infinity.
We briefly compare Theorem 3.4 with the convergence theorems of Buhmann and Powell, cf. [4, 5]. Theorem 4.6 of [5] states that if \(\kappa \notin {\mathbb {N}} \) and if \(f \in C^{\lceil \kappa \rceil } ({\mathbb {R}}^n ) \) such that
then \(|| s_h [f] - f ||_{\infty } \le C h^{\kappa } . \) If \(\kappa \) is an odd integer, Buhmann loc cit., Theorem 4.7, needs an additional degree of differentiability, with the derivatives of order \(\kappa + 1 \) again bounded. Note that these condititons imply that f is of polynomial growth of order at most \(\lceil \kappa \rceil - 1 . \) On the other hand, Theorem 3.4 covers cases when f can have stronger growth, and derivatives of order \(\lceil \kappa \rceil - 1 \) do not not need to be bounded, for example \(f(x) = (1 + |x | ^2 )^{p/2 } \) with \(\lfloor \kappa \rfloor< p < \kappa \) if \(\kappa \notin {\mathbb {N}} \) and \(\kappa - 1< p < \kappa \) if \(\kappa \in {\mathbb {N}}^* \). We also note that condition (20) implies that f is \(C^{\lfloor \kappa \rfloor } \) (a consequence of the integrability of \(\widehat{f } (\xi ) |\xi |^{\kappa } \) on \(|\xi | \ge 1 \) and the smoothness of the inverse Fourier transform of any compactly supported distribution) without the derivatives of order \(\lfloor \kappa \rfloor \) necessarily being bounded (because of the singularity of \(\widehat{f}\) at 0).
Remark 3.7
It can be shown that if \(\kappa \notin {\mathbb {N}} \) then Theorem 3.4 remains true if \(\widehat{f } | _{{\mathbb {R}}^n \setminus 0 } \) can be identified with a Borel measure \(\nu \) on \({\mathbb {R}}^n \setminus 0 \) for which \(|\xi |^{\kappa } \in L^1 ({\mathbb {R}}^n , d|\nu | ) \), with the Wiener norm in (21) replaced by the total variation norm of \(\Sigma (\widehat{f } ) - \widehat{f } \) (which can be shown to be as measure on \({\mathbb {R}}^n \)). This implies a uniform estimate
We next show that if one is satisfied with a slower rate of convergence, the decay of \(\widehat{f } (\xi ) \) at infinity can be weakened. This will be relevant for our discussion of the convergence of the semi-discrete RBF scheme in sections 5 and 6, where the rate of convergence will anyhow be smaller than \(\kappa \) if the order of the operator is positive. Recall the spaces \(L^1 _{r, s } ({\mathbb {R}}^n ) \) and their norm \(|| \cdot ||_{1; r, s } \) defined by (2).
Theorem 3.8
Suppose that \(0 < r \le \kappa \) and that \(f \in L^{\infty } _{ - p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that
Then for all \(h > 0 \),
The constant C can be taken the same as in Theorems 3.4 and 3.2.
Proof
The hypotheses on \(\widehat{f } \) certainly imply that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 \left( {\mathbb {R}}^n , \min (|\xi |^{\kappa } , 1 ) d\xi \right) \), so we can apply Lemma 3.5. In particular, the estimate (18) still holds. We now split this integral as a sum of integrals over \(h|\xi | \le h \), \(h \le h |\xi | \le 1 \) qnd \(h |\xi | \ge 1 \), and use the Fix - Strang condition in 0, together with the trivial bound \(|\eta |^{\kappa } \le |\eta |^r \) if \(|\eta | \le 1 \) to bound
\(\square \)
This theorem shows the familiar interplay between the order of convergence of RBF interpolation and the smoothness of the function which is interpolated, quantified here by the decay of \(\widehat{f }(\xi ) \) at infinity. See Buhmann and Dai [6, 7] for related results on convergence of quasi-interpolants in weighted Sobolev norms.
Remark 3.9
We end this section with some observations on the case of \(\kappa = 0 . \) Theorem 3.4 remains true but does not seem very informative. It does not imply that \(s_h [f ] \) converges to f and indeed it cannot: it is well-known that stationary RBF interpolation with for example Gaussians does not converge, and as we will show in the next section, the best rate of approximation with basis functions in \(\mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) is \(h^{\kappa } . \) However, as we will also see in the next section, we can still have approximate approximation, in the sense that for functions f whose Fourier transform decays sufficiently rapidly at infinity, \(\limsup _{h \rightarrow 0 } || s_h [f ] - f ||_A \) can be made arbitrarily small by an appropriate choice of basis function: see Theorem 4.1 and the discussion following it.
We should note two technical problems with interpolation with general Buhmann-class functions if \(\kappa = 0 \): Theorem 2.3 no longer guarantees that \(L_1 (x) \) decays sufficiently rapidly to be integrable, though it may do so for particular examples such as the Gaussian, or under stronger conditions on \(\varphi \): cf. Buhmann’s result on the decay of \(L_1 (x) \) when \(\kappa \) is an even integer, mentioned in Remark 2.4(ii). The function \(s_h [f ] \) is therefore no longer guaranteed to be in \(L^1 ({\mathbb {R}}^n ) \), even if \(f \in \mathcal {S } ({\mathbb {R}}^n ) \), but its Fourier transform will then still exist as a tempered distribution, and Lemma 3.1 will still be true: the sequence \(\sum _{|j | \le N } f(hj ) L_1 (h^{-1 } x - j ) \) converges to \(s_h [f ] \) in \(\mathcal {S } ' ({\mathbb {R}}^n ) \) and the Fourier transforms, \(( \sum _{|j | \le N } f(hj ) e^{ - i h (j ,\xi ) } ) h^{n } \widehat{L }_1 (h \xi ) \) converge to \(\Sigma _h (\widehat{f } ) \) by Poisson’s formula.
Next, if \(\kappa = 0 \), the defining series for \(s_h [f ] \) will not necessarily converge if \(\widehat{f } \in L^1 ({\mathbb {R}}^n ) \) unless we impose an additional condition such as \(f \in L^{\infty } _{\varepsilon } ({\mathbb {R}}^n ) \) for some \(\varepsilon > 0 \), as in Theorem 3.4 when \(\kappa = 0 . \) However, if we only know that \(\widehat{f } \) is integrable, the series for \(s_h [f ] \) will still be summable in the sense that if \(\chi \in \mathcal {S } ({\mathbb {R}}^n ) \) such that \(\chi (0) = 1 \) and if we let
then \(s_h ^{\varepsilon } [f ] \) converges uniformly as \(\varepsilon \rightarrow 0 \) to a continuous function whose Fourier transform is \(\Sigma _h (\widehat{f } ) \), and this independently of the choice of \(\chi . \) To show this, note that the Fourier transform of the right hand side is equal to \((2 \pi )^{-n } \Sigma _h (\widehat{\chi } _{\varepsilon } * \widehat{f } ) \), where \(\widehat{\chi } _{\varepsilon } (\xi ) = \varepsilon ^{-n } \widehat{\chi } (\varepsilon ^{-1 } \xi ) . \) Since \((2 \pi )^{-n } \widehat{\chi }_{\varepsilon } * \widehat{f } \rightarrow \chi (0) \widehat{f } = \widehat{f } \) in \(L^1 \) and since \(\Sigma _h \) is a contraction, it follows that \(s_h ^{\varepsilon } [f ] \) converges in Wiener norm, and therefore in sup-norm, to the inverse Fourier transform of the, integrable, function \(\Sigma _h (\widehat{f } ) . \) If we now define \(s_h[f ] \) as the limit of the \(s_h ^{\varepsilon } [f ] \), the estimate (18) follows as before. We in fact only need the Fourier transform of \(\chi \) to be integrable, and if \(n = 1 \) we can for example take \(\chi (x) = \max ( 1 - |x | , 0 ) \), which implies that the series for \(s_h [f] \) is Césaro summable.
4 Approximate Approximation
It is easy to show that the approximation error in the Wiener norm cannot in general go to 0 faster than \(h^{\kappa } \): if \(\widehat{f } \in L^1 ({\mathbb {R}}^n ) \) has compact support, then the supports of \(\widehat{f } (\cdot + 2 \pi k / h ) \) will be disjoint if h is sufficiently small. It follows that
since \(\sum _k \widehat{L }_1 (\eta + 2 \pi k ) = 1 . \) If we define
then Fatou’s lemma implies that
We will see below that \(\underline{l }_{\kappa } > 0 . \) The inequality (28) remains valid if \(\widehat{f } \) is not compactly supported but decays sufficiently fast at infinity: see Theorem 4.3 below. Here we first examine the corresponding upper bound.
As we just noted, one cannot in general do better that \(O(h^{\kappa }) \) for the approximation error. However, for suitable basis functions \(\varphi \) and for \(\widehat{f }(\xi ) \) which decay sufficiently fast at infinity we may observe a higher apparent rate of convergence for h’s which are small but not too small. If \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \), we letFootnote 1
A slight modification of the proof of Theorems 3.2 and 3.4 then gives the following more precise estimate for the approximation error.
Theorem 4.1
Let \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with non-negative order of singularity \(\kappa \ge 0 \) and decay rate \(N > n \), and suppose that \(f \in L^{\infty } _{- p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that for some \(s > \kappa \),
Then there exists for each \(\varepsilon > 0 \), a constant \(C_{\varepsilon } = C_{\varepsilon , \varphi } \) such that
Concretely, the condition on \(\widehat{f } \) means that \(\widehat{f } \) has to be integrable with respect to \(|\xi |^{\kappa } \) near 0 and with respect to \(|\xi |^s \) near infinity. In particular, f will be in \(C^{\lfloor s \rfloor } _b . \) The theorem then implies that if \(\overline{l }_{\kappa } (\varphi ) \) is very small, then the rate of convergence for small, but not too small h’s will at first appear to be \(h^s \ll h^{\kappa } \), up to the point that the first term of (30) dominates and the error saturates at a level comparable to \(\overline{l }_{\kappa } (\varphi ) h^{\kappa } . \) This is related to the notion of approximate approximation introduced by Maz’ya [24], see also Maz’ya and Schmidt [25, 26]. These authors examine quasi-interpolants \(\sum _j f(jh) \varphi _h (x - jh ) \) of a function f, with \(\varphi (x) \) rapidly decreasing (a Gaussian in [25]). Since we are in the case \(\kappa = 0 \), these quasi-interpolants are not expected to converge to f(x) , and indeed they will not, but it is shown in [26] that if \(\varphi (x) \) is taken of the form \(\phi (x/c ) \), where \(\phi \) is smooth, satisfies certain moment conditions and decays sufficiently rapidly at infinity, and if f has bounded derivatives of order L, then by choosing c sufficiently large one can achieve an apparent order of convergence of \(h^L \) modulo a saturation error which is non-zero but can be made arbitrarily small by letting c go to infinity. The parameter \(c > 0 \) is usually called a shape parameter in this context.
The results of Maz’ya and Schmidt correspond to the case \(\kappa = 0 \) of Theorem 4.1 once we have shown that we can construct basis functions with arbitrarily small \(\overline{l }_{\kappa } (\varphi ) \), which we will do by the same construction involving shape parameters: see Proposition 4.6 below. Note, though, that Theorem 4.1 treats the a priori more difficult case of exact interpolation instead of quasi-interpolation, although the proof turns out to be quite easy, given the preparatory work of the previous section. If \(\kappa > 0 \), as already noted, we have ordinary convergence but can have apparent convergence at a higher rate if f is sufficiently smooth.
We will encounter similar approximate approximation phenomena when studying convergence rates of RBF schemes in sections 6 and 7 below.
Proof of theorem 4.1
It suffices, by (18), to bound \(|| (1 - \widehat{L }_1 (h \xi ) ) \widehat{f }(\xi ) ||_1 . \) Let \(\overline{l } := \overline{l }_{\kappa } (\varphi ) . \) Then if \(\varepsilon > 0 \), there exists a \(\rho (\varepsilon ) > 0 \) such that if \(h |\xi | < \rho (\varepsilon ) \), then \(0 \le 1 - \widehat{L }_1 (h\xi ) \le \frac{1 }{2 } (1 + \varepsilon ) \overline{l } \cdot h^{\kappa } |\xi |^{\kappa } \), and
which implies the theorem. \(\square \)
Corollary 4.2
If \(\widehat{f } \) satisfies the conditions of Theorem 4.1, then
The next theorem complements this upper bound by the lower bound (28) when \(\widehat{f } \) is not necessarily compactly supported.
Theorem 4.3
Let f satisfy the hypothesis of Theorem 4.1: \(f \in L^{\infty } _{- p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) and \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 ({\mathbb {R}}^n , \max \left( |\xi |^{\kappa } , |\xi |^s ) \right) \) for some \(s > \kappa . \) Then
Proof
We only need to establish the lower bound. If \(|\xi |_{\infty } = \max _j |\xi _j | \) is the \(\ell ^{\infty } \)-norm on \({\mathbb {R}}^n \), let \(Q_h = \{ \xi \in {\mathbb {R}}^n : |\xi |_{\infty } \le \pi / h \} = [ - \pi / h , \pi / h ]^n \), the cube centered at 0 with sides \(2\pi / h \), and let \(Q_h (\ell ) =2\pi h^{-1 } \ell + Q_h . \) Then
so that
The double sum in the second line can be bounded by
where we used that
The first line of (32), on account of \(\widehat{L }_1 \) taking values in [0, 1] , equals
where \(\mathbf {1 }_{Q_h } \) is the indicator function of \(Q_h . \) Since \(\mathbf {1 }_{Q_h } \rightarrow 1 \) as \(h \rightarrow 0 \), the lower bound now follows once more by Fatou’s lemma and the definition of \(\underline{l }_{\kappa } (\varphi ) . \) \(\square \)
The following proposition gives an explicit formula for \(\underline{l }_{\kappa } \) and \(\overline{l }_{\kappa } : \)
Proposition 4.4
For \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with \(\kappa \ge 0 \) and \(N > n \), let
Then if \(\kappa > 0 \),
while if \(\kappa = 0 \),
Note that \(\underline{A }(\varphi ) > 0 \) by definition 2.1 (iii) and that the series in these formulas converges absolutely since \(N > n . \)
Proof
If \(L_1 = L_1 (\varphi ) \) is the Lagrange function associated to \(\varphi \), then
If we let \(R(\eta ) := \sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k ) \), then R is continuous (even \(C^{\lfloor \kappa \rfloor + n + 1 } \)) in a neighborhood of 0. Since
(34) and (35) follow upon letting \(\eta \rightarrow 0 . \) \(\square \)
Corollary 4.5
If \(\lim _{\eta \rightarrow 0 } |\eta |^{- \kappa } \widehat{\varphi } (\eta ) \) exists, then \(\underline{l }_{\kappa } (\varphi ) = \overline{l }_{\kappa } (\varphi ) = l_{\kappa } (\varphi ) \), and
for f as in Theorem 4.1 with \(s > \kappa . \)
One can often construct basis functions with small \(\overline{l } _{\kappa } (\varphi ) \) by introducing a so-called shape-parameter c and taking \(\varphi \) of the form \(\varphi (x) = \phi _c (x) := \phi (x/c ) \) with c large, for \(\phi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with N sufficiently large:
Proposition 4.6
Suppose that \(\phi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with singularity \(\kappa \ge 0 \) and decay rate \(N > \max (\kappa , n ) . \) Then \(\lim _{c \rightarrow \infty } \overline{l }_{\kappa } (\phi _c ) = 0 . \)
Proof
Since \(\widehat{\phi }_c (\eta ) = c^n \widehat{\phi } (c \eta ) \), it follows that \(\underline{A }(\phi _c ) = c^{n - \kappa } \underline{A }(\phi ) \), and therefore, by (34), if \(\kappa > 0 \),
which tends to 0 as \(c \rightarrow \infty \) under the stated conditions on \(\kappa . \) The case of \(\kappa = 0 \) also follows since
\(\square \)
Examples of basis functions \(\phi \) which satisfy the conditions of the corollary are the Gaussians (for which \(\kappa = 0 \)) and the generalized multiquadrics, whose Fourier transforms decay exponentially at infinity, but none of the polyharmonic basis functions, since for these \(\widehat{\phi }(\xi ) \) is homogeneous and therefore \(\kappa = N . \) See Examples 5.12 below for further discussion. In fact, for a Gaussian or a multiquadric, \(\widehat{\phi }(\xi ) \) decays expontially at infinity, and \(\overline{l }_{\kappa } (c) \) will decay exponentially in c.
4.1 Dependence of the Constants on the Shape Parameter
If we want to apply (30) with \(\varphi = \phi _c \) as above (and some fixed \(\varepsilon \)), it becomes interesting to ask how the constant \(C_{\varepsilon } = C_{\varepsilon , \varphi } \) depends on the shape-parameter c. One expects it to go to infinity with c, and the question then is at what rate exactly. We will see that under reasonable assumptions on \(\widehat{\phi } (\eta ) \), this constant behaves like \(c^s . \)
We first analyse its dependence on \(\varphi \) for a general \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) . \) From the proof, \(C_{\varepsilon , \varphi } \) is proportional to \(\rho (\varepsilon , \varphi ) ^{-s } \), where \(\rho (\varepsilon , \varphi ) \) is any positive number such that \(2 | \eta |^{- \kappa } (1 - \widehat{L }_1 (\eta ) ) < (1 + \varepsilon ) \overline{l }_{\kappa } \) for \(|\eta | \le \rho (\varepsilon ) . \) If we put
then
If we introduce the family of semi-norms
then \(| R(\eta ) - R(0 ) | \le p_r (|\nabla \widehat{\varphi } | ) \, |\eta | \) for \(|\eta | \le r \), with \(|\nabla \varphi | \) the euclidean norm of the gradient, so that \(0 \le 2 R(\eta ) < 2 (1 + \varepsilon ) R(0) \) if \(|\eta | \le \rho _1 \), where
On the other hand, there exists a \(\rho _2 := \rho _2 (\varepsilon ; \varphi ) \) such that \( | \eta |^{\kappa } \widehat{\varphi } (\eta ) \ge (1 - \varepsilon ) \underline{A }(\varphi ) \) if \(|\eta | \le \rho _2 \), where \(\underline{A } = \underline{A } (\varphi ) \) was defined in proposition 4.4 above. Combining these two estimates and observing that \(\overline{l }_{\kappa } = 2 R(0) / \underline{A } \) we see that
if \(|\eta | \le \rho \) where
and we get an approximate approximation estimate in the form
We now derive lower bounds for \(\rho \) when \(\varphi = \phi _c \), for a fixed \(\varepsilon \), which we will sometimes drop from the notations. First of all, there exists a \(r(\phi ) = r(\phi ; \varepsilon ) \) such that \(\widehat{\phi } (\eta ) \ge (1 - \varepsilon ) \underline{A } (\phi ) \) if \(|\eta | \le r(\phi ) . \) This implies that \(|\eta |^{\kappa } \widehat{\phi _c } (\eta ) = c^n |\eta |^{\kappa } \widehat{\phi } (c \eta ) \ge c^{n - \kappa } \frac{1 }{2 } \underline{A } (\phi ) = \frac{1 }{2 } \underline{A } (\phi _c ) \) if \(c |\eta | \le r(\phi ) \), so we can take \(\rho _2 (\varepsilon ; \phi _c ) = r(\phi ) / c . \)
The behavior of \(R_{\phi _c } (0) / p_{\pi , N }( \nabla {\widehat{\phi _c } } ) \) for large c depends on the asymptotic behavior at infinity of \(\widehat{\phi } \) and of \(\nabla \widehat{\phi } . \) We consider the case of a polynomially decaying \(\widehat{\phi } \) and that of an exponentially decaying one.
Example 4.7
(Polynomially decaying \(\widehat{\phi } \)) Suppose that
Then
On the one hand, we have for any \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) that
which implies that
which can certainly be bounded from above by a constant times \(c^{n + 1 - N } \) if \(c \ge 1 . \) Taking \(r = \pi \) in (38) we therefore find that \(\rho _1 (\varepsilon , \phi _c ) \ge C c^{-1 } \) for some constant C. It follows that \(\rho (\varepsilon , \phi _c ) \ge C c^{-1 } \) for some constant C and (40) implies an estimate
with constants \(C_1 \) and \(C_2 \) which depend on \(\phi \) (and on \(\varepsilon \)), but not on the shape parameter c. Neglecting the numerical values of these constants, the second term will dominate the first as long as \(h \gg c^{-1 - N/(s - \kappa ) } \), and the estimate will be relevant for h’s in the range \(c^{-1 - N/(s - \kappa ) } \ll h \ll c^{-1 } \) (since we want \( (ch)^s \ll 1 \)).
Example 4.8
(Exponentially decaying \(\widehat{\phi } \)) Suppose that \(\phi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) is such that there exists constants \(p \ge 0 \) and \(C > 0 \) for which
and
An example of such a \(\phi \) is given by the multi-quadric on \({\mathbb {R}}^n \): see Example 5.12 below.
Since \(\widehat{\phi _c } (\eta ) = c^n \widehat{\phi } (c \eta ) \), \(p_r (|\nabla \widehat{\phi _c } | ) \) will then be bounded by a constant times
Using that if \(|\eta | \le r \), \(e^{ - c | \eta + 2 \pi k | } \le e^{c r } e^{- 2 \pi c |k | } \) and, assuming wlog that \(r \le \pi \), that \(|\eta + 2 \pi k | \ge 2 \pi (|k | - \frac{1 }{2 } ) \ge \pi |k | \) for \(|k | \ge 1 \), we find that this expression is bounded by
and using the first inequality of (43), it follows that \(p_r (|\nabla \widehat{\phi _c } | ) \) is bounded by a constant times \(c R_{\phi _c } (0) . \) Remembering (38), it follows that \(\rho _1 (\phi _c ) \) is bounded from below by a constant times \(\min ( \pi , r , c^{-1 } e^{- c r } ) . \) This still has r as a free parameter, and taking \(r = c^{-1 } \) and remembering the estimate for \(\rho _2 (\phi _c ) \) from the previous example, we find that we can take \(\rho (\phi _c ) \ge C \cdot c^{-1 } . \) Since (43) implies that \(\underline{l }_{\kappa } \le C \cdot c^{\kappa - p } e^{- 2 \pi c } \), we then have the approximate approximation estimate
which is relevant for the range \(c^{- 1 - p / (s - \kappa ) } e^{- 2 \pi c / (s - \kappa ) } \ll h \ll c^{-1 } \), where \(c \gg 1 . \)
Example 4.9
(The Gaussian) A final interesting example is that of the Gaussian, with \(\widehat{\phi } (\eta ) = e^{- |\eta |^2 } . \) In this case, \(\nabla \widehat{\phi _c } (\eta ) = c^{n + 2 } \eta e^{- c^2 |\eta |^2 } \), and \(p_r (|\nabla \widehat{\phi _c } | ) \) for \(r \le \pi \) is bounded by a constant times
for any \(0< \epsilon < 1 \), where we used the inequality \((a - b )^2 \ge (1 - \epsilon ) a^2 - (\epsilon ^{-1 } - 1 ) b^2 \) with \(a = 2 \pi |k | \) and \(b = |\eta | . \) Asymptotically, for large c, this behaves like \(c^{n + 2 } e^{ - c^2 (1 - \epsilon ) 4\pi ^2 + (\epsilon ^{-1 } - 1 ) c^2 r^2 } \), while \(R_{\phi _c } (0) c^{n + 2 } \simeq e^{- 4 \pi ^2 c^2 } \), and we can choose \(\rho _1 (\phi _c ) \) (with a fixed \(\varepsilon \)) to be bounded from below by a constant times
If we now take the free parameters r and \(\epsilon \) equal to \(c^{- 2 } \) and remember that \(\rho _1 (\phi _c ) \simeq c^{-1 } \), we conclude that we can choose \(\rho (\phi _c ) \simeq c^{-2 } \), and we have the following approximate approximation estimate for Gaussian RBF interpolation (recalling that \(\kappa = 0 \) in this case):
where the constant \(C_1 \) can be taken arbitrarily close to 1.
5 Convergence of the Stationary RBF Scheme for the Heat Equation
5.1 RBF Scheme for the Heat Equation
We introduce an RBF scheme for the Cauchy problem for the classical heat equation,
\(\Delta = \sum _{j = 1 } ^n \partial _{x_j } ^2 \) being the Laplace operator, and examine its convergence. The scheme is a variant of the classical method of lines, and looks for approximate solutions \(u_h \) of the form
where the \(c_k (\cdot ; h ) : [0, \infty ) \rightarrow {\mathbb {R}} \) are differentiable functions. Here, \(L_1 \) is the Lagrange function of theorem 2.3, associated to a given basis function \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) which we fix. We will assume throughout this section that the decay rate \(N > n + 2 \) and that \(\kappa > 0 . \) We will see below that for the scheme to converge we need that \(\kappa > 2 \) while for \(\kappa = 2 \) we can have approximate convergence.
The coefficients \(c_k (t; h ) \) of \(u_h \) are determined by requiring that \(u_h \) solve (44) exactly in the points of \(h {\mathbb {Z}}^n \):
while \(u_h (x, 0 ) \) is taken to be equal to \(s_h [f ] (x) \), the RBF-interpolant of f. This leads to the following initial value problem for the coefficients \(c_j (t; h ) \):
Since this is an infinite system of ODEs we first discuss existence and uniqueness of solutions in suitable Banach spaces.
For \(s \in {\mathbb {R}} \), let
One easily verifies using Theorem 2.7 that the convolution operator
is a bounded operator on \(\ell ^{\infty } _{-p } \) if \(0 \le p < \kappa . \) Indeed,
using that \((1 + |k| ) \le (1 + |j - k | )( 1 + |j | ) . \) The sum of the series on the right is independent of j and finite if \(p < \kappa \), by Theorem 2.7.
It follows that if we let \(c(t) := (c_j (t) )_j \), and if the initial value \(c(0) \in \ell ^{\infty } _{- p } \) for some \(p \in [0 , \kappa ) \), the system (47) has a unique \(\ell ^{\infty } _{- p } \)-valued solution which is given by \(c(t) = e^{h^{-2 } t A_L } (c(0) ) . \) Next, if for \(c \in \ell ^{\infty } _{-p } \) we let (with some abuse of notation)
then \(s_h : c \rightarrow s_h [c] \) is a bounded linear operator from \(\ell ^{\infty } _{-p } \rightarrow L^{\infty } _{-p } ({\mathbb {R}} ) \) if \(0 \le p < \kappa . \) Indeed, using the decay of \(L_1 \),
The sum on the right converges and defines a 1-periodic continuous function on \({\mathbb {R}}^n \) which is therefore uniformly bounded, while the factor in front can be estimated by \(\max (1 , h^{-n } ) . \)
If \(f \in L^{\infty } _{-p } ({\mathbb {R}}^n ) \), we can in particular take \(c(0) = f |_{h {\mathbb {Z}}^n } \), and
is the unique function (45) whose coefficients satisfy (47). We summarize this discussion in the following lemma:
Lemma 5.1
If \(0 \le p < \kappa \) and if \(f \in L^{\infty } _{-p } ({\mathbb {R}}^n ) \) then there is a unique function \(u_h = u_h [f ] \in C^1 \left( [0 , \infty ) ; L^{\infty } _{-p } ({\mathbb {R}}^n ) \right) \) of the form (45) which satisfies (46) and \(f \rightarrow u_h [f ] (\cdot , t ) \) is a bounded linear map on \(L^{\infty } _{-p } ({\mathbb {R}} ) \), for each \(t \ge 0 \) and \(h > 0 . \)
In particular, for each fixed t, \(u_h (x, t ) \) has tempered growth in x, and thus possesses a well-defined Fourier transform, which we will study next.
5.2 Convergence in Wiener Norm
We start by computing the Fourier transform of \(u_h = u_h [f ] . \) Let us introduce the auxiliary function \(G (\eta ) \) on \({\mathbb {R}}^n \) by
where the series converges absolutely, since \(N > n + 2 . \)
Lemma 5.2
Let \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}} ) \) with decay rate \(N > n + 2 \) and \(\kappa > 0 . \) If \(\widehat{f } \in L^1 ({\mathbb {R}}^n ) \), then the Fourier transform of \(u_h (x, t ) \) with respect to x is given by
Proof
Since \(( \Delta L_1 (j ) )_{j \in {\mathbb {Z}}^n } \) is in \(\ell ^1 := \ell ^1 ({\mathbb {Z}}^n ) \), it follows that \(A_L \) is a bounded operator on \(\ell ^1 \), and hence the system (47) has an \(\ell ^1 \)-valued solution \(c(t) = (c_j (t) )_j \) if the initial value \(c(0) \in \ell _1 . \) In particular, if \(c(0) = f |_{h {\mathbb {Z}}^n } \) with \(f \in \mathcal {S } ({\mathbb {R}}^n ) \), then the function \((x, k ) \rightarrow c_k (t ; h ) L_1 ( h^{-1 } x - k ) \) is absolutely integrable on \({\mathbb {Z}}^n \times {\mathbb {R}}^n \) for each fixed \(t \ge 0 , h > 0 \), and an application of Fubini’s theorem shows that the Fourier transform of \(u_h (\cdot , t ) \) is given by \(\widehat{u }_h (\xi , t ) = h^n \widehat{L }_1 (h \xi ) \, \gamma _h (\xi , t ) \), where
the series being absolutely convergent. By (47),
where the second line follows from the Poisson summation formula, whose application is justified by the decay at infinity of \(\Delta L_1 \) and of its Fourier transform. Hence \(\partial _t \widehat{u }_h (\xi , t ) = - h^{-2 } G(h \xi ) \widehat{u }_h (\xi , t ) \) which, together with the initial condition \(u_h (x, 0 ) = s_h [f ](x) \) implies (50).
If \(\widehat{f } \in L^1 ({\mathbb {R}}^n ) \), (50) follows by a standard approximation argument: if \(\widehat{f }_{\nu } \rightarrow \widehat{f } \) in \(L^1 \) with \(f_{\nu } \) rapidly deceasing, then \(f_{\nu } \rightarrow f \) in \(L^{\infty } \), so by Lemma 5.1, \(u_h [f_{\nu } ] (\cdot , t ) \rightarrow u_h [f ] (\cdot , t ) \) in \(L^{\infty } \) also, since \(\kappa > 0 . \) Hence their Fourier transforms converge in \(\mathcal {S }' := \mathcal {S } ' ({\mathbb {R}}^n ) . \) On the other hand, \(\widehat{s_h [f_{\nu } } ] = \Sigma _h (\widehat{f_{\nu } } ) \rightarrow \Sigma _h (\widehat{f } ) = \widehat{s_h [f ] } \) in \(L^1 \) and since G is non-negative, \(e^{- h^{-2 } t G(h \xi ) } \widehat{s_h [f_{\nu } ] } (\xi ) \rightarrow e^{- h^{-2 } t G(h \xi ) } \widehat{s_h [f ] } (\xi ) \) in \(L^1 \), and therefore in \(\mathcal {S } ' .\) \(\square \)
The following proposition lists some useful properties of G.
Proposition 5.3
Suppose that \(\varphi \in \mathfrak {B }_{\kappa , N } \) with \(N > n + 2 \) and let \(G := G_{\varphi } \) be defined by (49). Then
(i) G is a positive \(2 \pi \)-periodic function, and \(G(\eta ) = 0 \) iff \(\eta \in 2\pi {\mathbb {Z}}^n . \)
(ii) There exists a constant \(C > 0 \) such that \(0 \le G(\eta ) - | \eta |^2 \le C | \eta |^{\kappa } \) for \(| \eta | \le \pi . \)
(iii) G belongs to the Hölder space \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) \) with \(\lambda = \kappa - (\lceil \kappa \rceil - 1 ) . \)
Proof
(i) The periodicity is obvious and the positivity of G is an immediate consequence of the positivity of \(\widehat{\varphi } . \) Next, \(G(\eta ) = 0 \) iff \(|\eta + 2\pi k |^2 \widehat{L }_1 (\eta + 2 \pi k ) = 0 \) for all k. Since \(\widehat{L }_1 \) is non-zero outside of \((2 \pi ) {\mathbb {Z}}^n \setminus 0 \), this implies that \(\eta \in 2 \pi \cdot {\mathbb {Z}}^n . \) Conversely, any such \(\eta \) is a zero, given that \(\widehat{L }_1 (2 \pi k ) = \delta _{0k } . \)
Assertion (ii) follows from
and the behaviour of \(\widehat{\varphi } (\eta ) \) for small \(\eta . \)
Finally, (iii) follows from Lemma 2.5. \(\square \)
Property (iii) allows us to extend Lemma 5.2 to functions of polynomial growth: compare Lemma 3.5.
Lemma 5.4
Suppose that \(f \in L^{\infty } _{-p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 \left( {\mathbb {R}}^n , (|\xi |^{\kappa } \wedge 1 ) d\xi \right) . \) Then the identity (50) holds in the sense of tempered distributions, for each \(t \ge 0 . \) Moreover, \(\widehat{u_h [f ] } (\cdot , t ) - \widehat{u } (\cdot , t ) \) can be identified with the function
which moreover is integrable on \({\mathbb {R}}^n . \)
Here, and below, \(\widehat{f } \) without argument will indicate the distribution, and \(\widehat{f }(\xi ) \) the function with which it can be identified on \({\mathbb {R}}^n \setminus 0 \); we recall that by Lemma 3.5, \(\widehat{s_h [f ] } - \widehat{f } \) can be identified with an \(L^1 \)-function. That the second term of (51) is in \(L^1 \) follows from Lemma 5.3(ii).
Proof
We just clarify the statement of the lemma, and refer to Appendix B for the proof, which uses elements of the proof of Lemma 3.5. The proof of that lemma shows that \(\widehat{s_h [f ] } = \Sigma _h (\widehat{f } ) \) extends to a continuous linear functional on \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) . \) Since (i) and (iii) of Proposition 5.3 imply that the function \(e^{- h^{-2 } t G(h \cdot ) } \) is in \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) \), its product with \(\Sigma _h (\widehat{f } ) \) is well-defined as an element of the dual of \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) . \) \(\square \)
We can now show convergence in Wiener norm of \(u_h \) to the solution of the Cauchy problem (44): recall again the spaces \({L }^1 _{r, s } \) defined in (2).
Theorem 5.5
Let \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with decay rate \(N > n + 2 \) and singularity of order \(\kappa \ge 2 \) and suppose that \(f \in L^{\infty } _{-p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that
Let \(u_h := u_h [f ] \) and let u be the solution to the Cauchy problem (44) with initial value f. Then there exists a constant \(C = C_{\varphi } \) independent of h and f such that for \(0 < h \le 1 \),
In particular, \(u_h \) converges to u in sup-norm at a rate of \(h^{\kappa - 2 } . \)
Proof
Note that the conditions on f are weaker than those of Theorem 3.4. Indeed, we will be applying Theorem 3.8 with \(r = \kappa - 2 . \) By lemma 5.4, we can estimate \(|| \widehat{u }_h (\xi , t ) - \widehat{u }(\xi , t ) ||_1 \) by
Since G is positive, the first term can be bounded by \( | | \, \widehat{s_h [f] } - \widehat{f } \, | |_1 \le C || \widehat{f } ||_{1; \kappa - 2 , \kappa } h^{\kappa - 2 } \), by Theorem 3.8. To estimate the second term, we first use Proposition 5.3(ii) together with the inequality \((1 - e^{-x } ) \le x \) for \(x \ge 0 \) to estimate the integral over \( h | \xi | \le \pi \) by
assuming wlog that \(h \le \pi . \) Since the integral over \(|\xi | \ge h^{-1 } \) can be bounded by
the theorem follows, where in fact we have established the more precise bound
with \(C_{1, \varphi } := \sup _{0 < |\eta | \le \pi } (G(\eta ) - |\eta |^2 ) / |\eta |^{\kappa } \) and \(C_{2, \varphi } := \sup _{0 < |\eta | } (1 - \widehat{L }_1 (\eta ) ) / |\eta |^{\kappa } \), the constant of Corollary 3.8. \(\square \)
Remarks 5.6
(i) If we strengthen the hypothesis on \(\widehat{f } \) to \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{\kappa - 2 } ({\mathbb {R}}^n ) \), we obtain the error bound \(C h^{\kappa - 2 } || \widehat{f } || _{1; \kappa - 2 } \) with a constant C which is independent of t.
(ii) The estimate for the integral over \(|\xi | \ge \pi / h \) is quite rough, but note that since \(h^{-2 } G(h\xi ) \) is \(2 \pi / h \)-periodic and equal to 0 in points of \(2 \pi h^{-1 } {\mathbb {Z}}^n \), \(e^{ - h^{-2 } tG (h\xi ) } - e^{ - t |\xi |^2 } \) can get arbitrarily close to 1 on this set. A similar remark applies to the first term of (53). There may be room for improvement, by further analyzing the contribution of a small neighborhood of this set of points.
It is not difficult to verify that \(h^{\kappa - 2 } \) is the exact order of approximation if \(\kappa > 2 \), and that the scheme does not converge if \(\kappa = 2 . \) Let
We will see in Proposition 5.10 below that \(\underline{g }_{\kappa } > 0 . \)
Theorem 5.7
Let the basis function \(\varphi \) be as in Theorem 5.5 and let \(f \in L^{\infty } _{-p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{k, \kappa } ({\mathbb {R}}^n ) \) for some \(k \in ( \kappa - 2 , \kappa ] . \) Then if \(\kappa > 2 \),
while if \(\kappa = 2 \),
Proof
Since \(|| \widehat{s_h [f] } - \widehat{f } ||_1 = O (h^k ) \) and \(k > \kappa - 2 \), Lemma 5.4, Theorem 3.8 and Fatou’s lemma imply that
where we have put \(R ( \eta )= G(\eta ) - |\eta |^2 . \) By the mean value theorem applied to the exponential, there exist \(\zeta = \zeta ( h , \xi , t ) \in [0, 1 ] \) such that
If \(\kappa > 2 \), then \(h^{-2}R( h \xi ) \rightarrow 0 \) as \(h \rightarrow 0 \) by Proposition 5.3(ii), and (56) follows from
If \(\kappa = 2 \), then
which proves (57). \(\square \)
5.3 Approximate Approximation Properties of the Scheme
As we have just seen, our RBF scheme for the heat equation does not converge if \(\kappa = 2 \), which is for example the case if \(n = 1 \) and the basis function is the Hardy multiquadric. It turns out that we can then still achieve an arbitrarily small absolute error by an appropriate choice of the basis function, e.g. by introducing a shape parameter. This is again an approximate approximation phenomenon of the type encountered in section 4, and which if \(\kappa > 2 \) will again take the form of an apparent rate of convergence better than than \(O( h^{\kappa - 2 } ) \), for h’s which are small but not too small, up to some threshold \(h_0 \), and for initial conditions f whose Fourier transform decay sufficiently rapidly at infinity. Suppose again that \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with decay rate \(N > n + 2 \) and singularity of order \(\kappa \ge 2 \), and let the initial condition f be as in Lemma 5.4. Let
and recall the Definition (29) of \(\overline{l }_{\kappa } . \)
Theorem 5.8
Let \(\kappa > 2 \) and \(s > \kappa - 2 . \) If \(s > \kappa \) then there exist for any \(\varepsilon > 0 \) a constant \(C_{\varepsilon } \) which does not depend on \(t \ge 0 \) such that if f has polynomial growth of order strictly less than \(\kappa \) with \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{\kappa } \cap L^1 _s \), then for sufficiently small \(h > 0 \),
while if \(\kappa - 2 < s \le \kappa \) and \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{s , \kappa } \), then
If \(\kappa = 2 \), we can replace the first term on the right in these inequalities by
Proof
We adapt the proof of theorem 5.5. If \(\varepsilon > 0 \), then since \(\overline{g}_{\kappa } > 0\) exists a \(r(\varepsilon ) > 0 \) such that \( | G(\eta ) - | \eta |^2 | \le (1 + \varepsilon ) \overline{g }_{\kappa } | \eta |^{\kappa } \) if \( | \eta | < r(\varepsilon ) . \) Hence, assuming wlog that \(r(\varepsilon ) \le \pi \),
We estimate the integral over \(|\xi | \ge r (\varepsilon ) / h \) by \(2r (\varepsilon ) ^{-s } h^{-s } \int _{|\xi | \ge r(\varepsilon ) / h } \, |\widehat{f } (\xi ) | \, |\xi |^s d\xi \), which we bound by \(2 r(\varepsilon )^{-s } h^s || \widehat{f } || _{1; s } \) if \(s > \kappa \), and by \(|| \widehat{f } || _{1; s , \kappa } \) if \(s \le \kappa \), assuming in the latter case that \(h \le r (\varepsilon ) . \) If we finally use Theorem 4.1 to estimate first term of (53) when \(s > \kappa \), and Theorem 3.8 when \(s \le \kappa \), the theorem follows. \(\square \)
Corollary 5.9
Let \(\kappa \ge 2 \) and suppose that \(g_{\kappa } := \lim _{\eta \rightarrow 0 } |G(\eta ) - |\eta |^2 | / |\eta |^{\kappa } \) exists, so that \(\underline{g }_{\kappa } = \overline{g }_{\kappa } =: g_{\kappa } . \) Suppose that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{s , \kappa } ({\mathbb {R}}^n ) \) for some \(s \in (\kappa - 2 , \kappa ] . \) Then
Compare with Corollary 4.5. It follows from Proposition 5.10 below that \(g_{\kappa } \) exists iff \(\lim _{|\eta | \rightarrow 0 } |\eta |^{\kappa } \widehat{\varphi }(\eta ) \) exists. One can also give a direct proof of this corollary using Lebesgue’s dominated convergence theorem: see the proof of theorem 6.8 below.
If \(\overline{g }_{\kappa } \) and \(\overline{l }_{\kappa } \) are small, then Theorem 5.8 with any fixed \(\varepsilon \), \(\varepsilon = 1 \) say, shows one may observe a higher apparent rate of convergence than the actual rate for small but not too small h’s if \(\widehat{f }(\xi ) \) decays sufficiently rapidly at infinity: compare our discussion of approximate approximation after Theorem 4.1. As in section 4, we can construct basis functions \(\varphi \) with small \(g_{\kappa } (\varphi ) \) by taking these of the form \(\varphi (x) = \phi (c^{-1 } x ) \) and letting \(c \rightarrow \infty . \) We start by deriving explicit formulas for \(\overline{g }_{\kappa } (\varphi ) \) and \(\underline{g }_{\kappa } (\varphi ) . \) Recall the definition of \(\underline{A } (\varphi ) \) and \(\overline{A } (\varphi ) \) in Proposition 4.4.
Proposition 5.10
We have
Proof
where \(g(\eta ) := \sum _{k \ne 0 } \left( 4 \pi (\eta , k ) + 4 \pi ^2 |k |^2 \right) \widehat{\varphi } (\eta + 2 \pi k ) \) and \(h(\eta ) := \sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k ) \) are continuous in a neighborhood of 0 . It follows that
with \(\underline{A } = \underline{A } (\varphi ) . \) The formula for \(\underline{g }_{\kappa } (\varphi ) \) follows similarly. \(\Box \)
Note that \(\overline{l }_{\kappa } (\varphi ) \le 2 \overline{g }_{\kappa } (\varphi ) . \) Also note that \(\underline{g }_{\kappa } (\varphi ) > 0 \) since \(\overline{A } < \infty \) and \(\overline{g }_{\kappa } (\varphi ) < \infty \) since \(\underline{A } > 0 \), by the ellipticity condition on \(\widehat{\varphi } \) at 0.
If we take \(\varphi (x) := \phi _c (x) = \phi (x/c ) \), with \(\phi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \), then \(\widehat{\varphi }(\eta ) = c^n \widehat{\phi }(c \eta ) \), and \(\underline{A }(\phi _c ) = c^{n - \kappa } \underline{A }(\phi ) . \) It follows that
where the series converges since \(N > n + 2 . \)
Corollary 5.11
If \(\phi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with decay rate \(N > \max (n + 2 , \kappa ) \), then \(\overline{g }_{\kappa } (\phi _c ) \rightarrow 0 \) as \(c \rightarrow \infty . \)
Examples 5.12
(i) Hardy’s multiquadric with shape parameter c is defined as
where the minus sign serves to make \(\widehat{\varphi } (\eta ) \) positive. Note that \( \varphi (x) = c \phi (x/c ) \) with \(\phi (x) := - \sqrt{|x |^2 + 1 } \), so that we are in the situation of corollary 5.11, except for an irrelevant multiplicative factor of c. The Fourier transform of \(\varphi \) on \({\mathbb {R}}^n \setminus 0 \) is given by
where \(K_{\nu } \) is the MacDonald function, or modified Bessel function of the second kind: see for example Baxter [2]. The limiting form of \(K_{\nu } \) for small values of the argument implies that as \(\eta \rightarrow 0 \), \(\widehat{\varphi }(\eta ) \simeq A_n |\eta |^{- n - 1 } \) with \(A_n = 2^{n } \pi ^{(n - 1 )/2 } \Gamma \left( \frac{n + 1 }{2 } \right) \), so that \(\kappa = n + 1 \), and our RBF-scheme for the heat equation will converge if \(n \ge 2 \), at a rate of \(h^{n - 1 } . \) The MacDonald functions are known to decay exponentially at infinity, \(K_{\nu } (z) \simeq (\pi / 2z )^{1/2 } e^{-z } \), so that we can conclude from (the proofs of) propositions 4.6 and 5.10 that \(\overline{l }_{\kappa } (\phi _c ) \) and \(\overline{g }_{\kappa } (\phi _c ) \rightarrow 0 \) at an exponential rate as \(c \rightarrow \infty . \)
If \(n =1 \), then \(\kappa = 2 \), and the scheme will not converge. However, Corollary 5.11 together with Theorem 5.8 shows that we can make the error arbitrarily small by taking the shape parameter c sufficiently large, with moreover an arbitrarily large apparent order of convergence for small but not-too-small h’s if the Fourier transform of the initial value decays sufficiently rapidly at infinity. At first sight, this may seem strange, because we are after all simply performing an additional scaling by c, and we are already using scaled basis functions \(\varphi _h (x) = \varphi (h^{-1 } x ) \) for our interpolation. Note, though, that we are interpolating with \(\phi _{ch } \) on \(h {\mathbb {Z}}^n \), and not on \(ch {\mathbb {Z}}^n . \) One can perform an analysis of the c-dependence of the constant \(C_{\varepsilon } \) in (59) (for any fixed \(\varepsilon > 0 \)) similar to the one of Sect. 4.1, with similar conclusions; we skip the details.
(ii) If we take a homogeneous basis function, \(\phi (x) = |x |^p \), \(p > 0 \), \(p \notin 2 {\mathbb {N}} \), or if \(\phi (x) = |x |^p \log |x | \) with \(p \in 2 {\mathbb {N}} \), then \(\widehat{\phi } (\eta ) \) is proportional to \(|\eta |^{- p - n } \) on \({\mathbb {R}}^n \setminus 0 \), so that \(\kappa = n + p = N \) and Corollary 5.11 does not apply, as indeed it shouldn’t: if \(\phi \) is homogeneous, then \(\widehat{L }_1 (\phi _c ) \) and \(G(\phi _c ) \) are independent of c, as are \(\overline{g }_{\kappa } (\phi _c ) \) and \(\overline{l }_{\kappa } (\phi _c ) \). We also note that for homogeneous basis functions \(\varphi (x) \) there is no difference between stationary and non-stationary interpolation, and our results directly apply to the latter. There then is an interesting dimensional effect: since \(\kappa = p + n \) increases with n, the RBF-scheme converges faster in higher dimensions. Such an effect was observed empirically in Chen et al. [11] for the example of the Dirichlet problem for the Laplacian on on \([0, 1 ]^n \) for \(n = 2, 3 \) and 4, solved by RBF collocation with \(\phi (x) = |x |^5 \) and uniformly distributed interpolation centers. They remark that, although known for stationary interpolation of functions, the observed increase of the convergence rate with space dimension when solving PDEs has no theoretical explanation yet. Theorems 5.5 and 6.4 below provide such an explanation, at least for translation-invariant evolution equations on \({\mathbb {R}}^n \) solved on regular grids, though further work remains to be done to extend this to mixed initial-boundary value problems solved using non-regular grids.
6 Convergence of Stationary RBF Schemes for Pseudo-differential Evolution Equations
The results of the previous section remain valid for a large class of evolution equations
with a generator a(D) which is a constant coefficient pseudo-differential operator (or Fourier multiplier operator) defined by \(\widehat{a(D ) u } = a \widehat{u } \), \(a = a(\xi ) \) being a function on Fourier space called the symbol of the operator. General pseudo-differential operators a(x, D) have symbols \(a (x, \xi ) \) which also depend on x, but these are outside of the scope of the present paper. However, the class of Fourier multiplier operators already contains many interesting examples which are relevant for applications in both physics and finance, notably the fractional Laplacians and the generators of Lévy processes. The latter have been treated numerically in [3] using the RBF scheme we investigate here, with good results.
Natural symbol classes for these Fourier multiplier operators are
where \(q \in {\mathbb {R}} \), \(\rho \ge 0 \), and \(\alpha \) ranges over all multi-indices \((\alpha _1 , \ldots \alpha _n ) \in {\mathbb {N}}^n . \) Pseudo-differential operators usually have symbols in \(S^q _1 \) (possibly allowing a singularity in 0, see below), but we will not need the improved decay for the derivatives, and we will show that the RBF-scheme for (62) convergences at a rate of \(h^{\kappa - q } \) if the symbol \(a \in S^q _0 \) and if the basis function \(\varphi \) is Buhmann class with a \(\kappa \ge q \), for initial values f in \(L^{\infty } _{- \kappa + \varepsilon } \) whose distributional Fourier transform is integrable with respect to \(|\xi |^{\kappa } \) (this can be weakened if the symbol a is elliptic).
Note that the operators a(D) are in general non-local. They can be thought of as singular integral operators since they are given by convolution with the inverse (distributional) Fourier transform of a which will have a singularity at the origin. The fact that we are dealing with integral operators might at first sight be thought to cause problems for the convergence of the RBF-scheme, since for stationary interpolation we need basis functions whose Fourier transform is singular in 0: they will therefore typically have polynomially growth, and will not be well localized. We note, though, that if \(a \in S^q _{\rho } \) for some \(\rho > 0 \), then the kernel of a(D) is known to be rapidly decaying off the diagonal.
Examples of operators a(D) with symbols in \(S^q _0 \) are the generators of (multi-dimensional) Lévy-processes whose Lévy measures have moments of all order: see Example 6.11 (i) below. These are in \(S^q _0 \) but in general not in \(S^q _{\rho } \) for a \(\rho > 0 . \) Simpler examples of such symbols are \(|\xi |^2 \cos |\xi | \), with a smooth cut-off near 0.
Another important class of operators for applications is that of the fractional Laplacians \((- \Delta )^s \) for \(s \in (0, 1 ) \), whose symbol \(|\xi |^{2s } \) is not in \(S^{2s } _0 \) since it is not smooth near 0. It is in the closely related symbol class \(\mathring{S }^{2s } _1 \), where
note that a larger \(\rho \) now means allowing a more singular behavior at 0 as well as requiring a more rapid decay at infinity for the derivatives. We will show that the RBF-scheme will still converge at a rate of \(h^{\kappa - q } \), provided we further restrict the initial condition f. That some restriction on f will be necessary can already be seen in the case of the fractional Laplacian, for which
with \(c_{n, s } |y |^{-n - 2s } \) the Fourier transform of \(|\xi |^{2s } \): see for example [21]. For the integral to converge at infinity we need restrictions on the growth of f, for example that \(f \in L^{\infty } _{- 2s + \varepsilon } \) for some \(\varepsilon > 0 . \) (The convergence at \(y = 0 \) requires some smoothness of f, for example a suitable (s-dependent) Hölder condition on f if \(s < 1/2 \), or on its derivative if \(1/2 \le s < 1 . \))
So as not to interrupt the flow of the arguments with this technicality, we will first examine convergence and approximate convergence of the RBF scheme for symbols in \(S^q _0 \) and then indicate in Sect. 6.5 the modifications which are necessary to accomodate symbols in \(\mathring{S }^q _1 . \) In the end our analysis will therefore apply to symbols in \(S^q _0 + \mathring{S }^q _1 = \left\{ a \in C^{\infty } ({\mathbb {R}}^n \setminus 0 ) : | \partial ^{\alpha } _{\xi } a (\xi ) | \le C_{\alpha } |\xi |^q \min (|\xi |, 1 ) ^{- |\alpha | } \right\} . \)
An obvious condition we will put on the symbol is that it has a non-negative real part,
This ensures that the initial value problem (62) is well-posed in the weighted Wiener spaces we will consider (as well as in numerous other function spaces, of course). It would have sufficed to require (64) only for \(\xi \) outside of a compact set, but we will for simplicity suppose that it holds globally. We will not need \(\mathrm{Re } \, a(\xi ) \) or \(a(\xi ) \) to be elliptic. In particular, our results will for example apply to the free Schrödinger equation, for which \(a(\xi ) = i |\xi |^2 \), or the so-called half-wave equation, with \(a(\xi ) = |\xi | . \) However, if \(a(\xi ) \) is elliptic, we will only need to impose a lower degree of smoothness (in the form of the rate of decay of the Fourier transform) on the initial condition f to have the algorithm convergence than when the symbol is non-elliptic.
We begin the detailed analysis of the RBF scheme for (62). Since the proofs will be similar to those for the heat equation in Sect. 5, we will only signal major differences. We again want to solve (62) using a semi-discrete stationary RBF-scheme, looking for approximate solutions of the form (45), where \(L_1 \) is the Lagrange function on \({\mathbb {Z}}^n \) associated to a basis function \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \), \(\kappa \ge 0 . \) We will now need that the decay rate \(N > n + q \): see Theorem 6.1. The coefficients \(c_k (t; h ) \) of \(u_h \) are once more determined by requiring that \(u_h \) solve (62) exactly in the interpolation points: \(\partial _t u_h (hj , t ) = - a(D) u_h (jh , t ) \) for \(j \in {\mathbb {Z}}^n \), while interpolating the initial value f at \(t = 0 . \) Since a(D) commutes with translations and since \(a(D) [L_1 (\cdot / h ) ] (x) = a (h^{-1 } D ) [L_1 ] (x/h ) \), where \(a(h^{-1 } D_x ) \) has symbol \(a(h^{-1 } \xi ) \), we now find that the coefficients \(c_k (t; h ) \) now have to satisfy the (infinite) system of ODEs:
with initial condition \(c_j (0; h ) = f(hj ) . \)
We first examine the action of a(D) on \(L_1 \):
Theorem 6.1
Let \(\kappa \ge 0 . \) If \(a \in S^q _0 ({\mathbb {R}}^n ) \) is a symbol of order q and if \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) where the decay rate \(N > n + q \), then \(a(D ) L_1 \) is a bounded continuous function and there exists a constant \(C > 0 \) such that \(| a(D ) L_1 (x) | \le C (1 + |x | )^{- (n + \kappa ) } . \)
See Appendix A, Remark A.3 for the proof. We note in passing that here, and for the other results below, it would have sufficed to require that \(|\partial ^{\alpha } _{\xi } a | \le C_{\alpha } |\xi |^q \) only for \(|\alpha | \le \lceil \kappa \rceil + n + 1 . \)
We suppose from now on that \(N > n + q . \) One shows as before (cf. Lemma 5.1) that for \(p < \kappa \) the system (65) has a unique solution in \(C^{\infty } ([0, \infty ) , \ell ^{\infty } _{ - p } ) \) for intial values in \(\ell ^{\infty } _{ - p } \) and that, as a consequence, \(u_h [f ] (\cdot , t ) \) is in \(L^{\infty } _{-p } ({\mathbb {R}}^n ) \) if \(f \in L^{\infty } _{- p } ({\mathbb {R}}^n ) \), with norm bounded by a constant times that of f.
Remark 6.2
A noteworthy feature of the RBF-scheme is that, contrary to for example Finite Difference schemes, we do not need to discretize the operator a(D) , but only need to know its action on \(L_1 \) or, equivalently, its action on \(\varphi \), if we implement the trial solution as \(\sum _k c_k (t) \varphi (h^{-1 } x - k ) . \) This is an advantage when the operator is a singular integral operator, as is typically the case for generators of Lévy processes: see [3] for concrete examples of such implementations and further discussion.
To analyze convergence of the RBF-scheme we introduce the auxiliary function \(G_a ^* \) on \({\mathbb {R}}^n \times {\mathbb {R}}_{> 0 } \) defined by
where the series converges absolutely since \(N > n + q . \) To make the connection with the previous section note that if \(a(\xi ) = |\xi |^2 \) then \(G_a ^* (\xi ; h ) = h^{-2 } G (h\xi ) \), with G given by (49).
One shows, similarly to Lemma 5.2, that if the initial condition f is a Schwarz-class function, or more generally if \(\widehat{f } \in L^1 ({\mathbb {R}}^n ) \), and if \(a \in S^q _0 \) satisfies (64), then the Fourier transform of \(u_h (x , t ) \) with respect to x is given by
Since \(G_a ^* (\xi ; h ) \) is in \(C_b ^{\lceil \kappa \rceil - 1 , \kappa - (\lceil \kappa \rceil - 1 ) } ({\mathbb {R}}^n ) \) (by the Fix-Strang estimates or, equivalently, the behaviour of \(\widehat{\varphi } (\eta ) \) near 0), we can extend this formula to initial values f with polynomial growth of order strictly less than \(\kappa \) whose Fourier transform coincides on \({\mathbb {R}}^n \setminus 0 \) with an element of \(L^1 ({\mathbb {R}}^n , (|\xi |^{\kappa }\wedge 1 ) d\xi ) \): see Appendix B. We also note that \(G_a ^* (\xi ; h ) \) is \(2\pi / h \)-periodic in \(\xi \) and has non-negative real part. Its zero-set contains \(2\pi h^{-1 } {\mathbb {Z}} \setminus 0 \) but may be bigger. We have the following basic estimate which generalizes proposition 5.3(ii).
Proposition 6.3
Suppose that \(a \in S^q _0 ({\mathbb {R}}^n ) \) for some \(q \in {\mathbb {R}} \), and that \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) where the decay rate \(N > n + q . \) Then there exists a constant C such that for all positive \(h < 1 \),
Proof
We have
The first term is bounded by a constant times \((1 + |\xi | )^q |h \xi |^{\kappa } \le C h^{\kappa - \max (q , 0 ) } |\xi |^{\kappa } \) if \(|h \xi | \le \pi . \) As for the other terms, \(|\xi + 2 \pi k / h | \) is comparable to |k|/h if \(|\xi | \le \pi / h \), so \(|a(\xi + 2 \pi k / h ) | \le C h^{-q } |k |^q \), and by (12), \( \widehat{L }_1 (h \xi + 2 \pi k ) \le C |h \xi |^{\kappa } |k |^{-N } \) for \(|h \xi | \le \pi \) and \(k \ne 0 \), so that
\(\square \)
6.1 Convergence
We will suppose from now on that \(f \in L^{\infty } _{ -p } ({\mathbb {R}}^n ) \) for some \(p < \kappa \) such that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 ({\mathbb {R}}^n , (|\xi |^{\kappa } \wedge 1 ) d\xi ) . \) The unique solution of the initial value problem in the space of tempered distributions is given by \(\widehat{u } (\xi , t ) = e^{- t a(\xi ) } \widehat{f } \) which is well-defined as a product of a tempered distribution and a \(C^{\infty } \)-function all of whose derivatives have at most polynomial growth. One shows, using the arguments of Appendix B, that \(u_h (x, t ) \) is a continuous function of polynomial growth of order at most \(\kappa \) for each \(t > 0 \), and that the Fourier transform of \(u_h (x , t ) - u(x, t ) \) is given by
where \(\widehat{s_h [f ] } - \widehat{f } \) identifies with the \(L^1 \)-function (104) and where \(\widehat{f } (\xi ) \) is the restriction of \(\widehat{f } \) to \({\mathbb {R}}^n \setminus 0 . \) We then have the following convergence theorem.
Theorem 6.4
Let \(a \in S^q _0 ({\mathbb {R}}^n ) \) be a symbol of order q with \(\mathrm{Re } \, a(\xi ) \ge 0 \), and suppose that \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) where the order of the singularity \(\kappa \ge q \) and the decay rate \(N > n + q . \) Then there exists a constant C such that if f is a function of polynomial growth of order strictly less than \(\kappa \) such that \(\widehat{f }|_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{\kappa } ({\mathbb {R}}^n ) \), then
Proof
The proof is similar to the proof of theorem 5.5, with a small twist. By Proposition 6.3 and the elementary inequality \(|e^z - 1 | \le |z | e^{\max ( \mathrm{Re } \, z , 0 ) } \) for \(z \in {\mathbb {C}} \), and sinceFootnote 2
for sufficiently small \(h |\xi | \), there exists an \(\rho > 0 \) such that if \( h |\xi | \le \rho \), then
which in absence of further hypotheses on the symbol \(a(\xi ) \) we simply bound by \(C h^{\kappa - \max (q, 0 ) } |\xi |^{\kappa } . \) The rest of the proof proceeds as before: since \(\mathrm{Re } \, G_a ^* \ge 0 \),
The last term can be estimated using Theorem 3.4 while the integral can be bounded by
Now use (71) for the first integral, and bound the second one by \((h / \rho )^{\kappa } \int _{|\xi | \ge \rho / h }|\xi | ^{\kappa } |\widehat{f } | \, d\xi \le (h / \rho )^{\kappa } || \widehat{f } || _{1; \kappa } . \) \(\square \)
Note that the estimate does not get better than \(O(h^{\kappa } ) \) if a is of negative order: \(q < 0 . \) We will show below that \(h^{\max ( \kappa - q , 0 ) } \) is the exact order of approximation if \(\mathrm{Re } (a ) \ne 0 \): see Theorem 6.8 and Remark 6.9 below.
Also note that, on comparing Theorem 6.4 for \(a(\xi ) = |\xi |^2 \) with Theorem 5.5, we require a stronger decay of \(\widehat{f } \) at infinity, which translates into two additional degrees of smoothnes (two extra derivatives) of f. If we assume that \(\mathrm{Re } \, a(\xi ) \) is elliptic, then
is independent of t and Theorem 6.4 remains valid if \(\widehat{f }|_{{\mathbb {R}}^n \setminus 0 } \in L^1 ({\mathbb {R}}^n , |\xi |^{\kappa - q } \wedge |\xi |^{\kappa } ) \), on replacing \(|| \widehat{f } || _{1; \kappa } \) by \(|| \widehat{f } || _{1; \kappa - q , \kappa } \): cf. the proof of theorem 5.5.
6.2 Approximate Approximation 1
The argument can be refined to give a rough approximate approximation estimate. The proof of theorem 6.4 shows that if \(r \le \min (\rho , \pi ) \) and if \(s > \kappa - q \ge 0 \) then
where
On closer examination, the proof of proposition 6.3 shows that
where we recall that \(|| a ||_{\infty , -q } = \sup _{\xi } (1 + |\xi |)^{-q } |a (\xi ) | \) (the constant \((4 \pi )^n \) in front is not optimal). It easily follows that
where we have put
with \(\underline{A } \) defined by (33); note that the series converges since \(N > n + q . \)
Also note that \(2 \overline{l }_{\kappa , 0 } (\varphi ) = \overline{l }_{\kappa } (\varphi ) = \overline{l }_{\kappa } . \) Equations (73) , (75) and Theorems 4.1 (when \(s > \kappa \)) and 3.4 (when \(\kappa - q < s \le \kappa \)) now imply the following rough approximate approximation estimate:
Theorem 6.5
Let the symbol a and the basis function \(\varphi \) be as in Theorem 6.4 with \(q \ge 0 . \) Then there exists for each \(\varepsilon > 0 \) a constant \(C_{\varepsilon } \) such that if \(s > \kappa - q \) and f has polynomial growth of order less than \(\kappa \) with \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{\kappa } ({\mathbb {R}}^n ) \cap L^1 _s ({\mathbb {R}}^n ) \)then
with \(C_a = (4 \pi )^q || a ||_{S^q _0 } . \)
The second term can obviously be absorbed in the final one if \(s \le \kappa . \) We call this a rough approximate approximation estimate since the bound does not correctly reflect the a-dependence. In particular, there is no similar lower bound for the approximation error. It does however imply that if \(\overline{l }_{q, \kappa } (\varphi ) \) is small (and therefore \(l_{\kappa } (\varphi ) \) also), then the error will appear to be \(O(h^s ) \) for small but not-too-small \(h > 0 . \) Theorem 6.5 can be applied with basis functions \(\varphi = \phi _c \) depending on a shape parameter c, since one easily shows that \(\overline{l }_{\kappa , q } (\phi _c ) \rightarrow 0 \) as \(c \rightarrow \infty \) if \(\phi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with decay rate \(N > \max (n + q , \kappa ) \): cf. Corollary 5.11 and its proof. One can also show that the constant \(C_{\varepsilon } \) in (76) (for any fixed \(\varepsilon > 0 \)) again behaves like \(c^s \), for the \(\phi \)’s we considered in Sect. 4.1.
6.3 Exact Order of the Scheme
To obtain a more precise result, and in particular an asymptotic lower bound for the approximation error, we introduce an additional hypothesis on a.
Definition 6.6
We will say that a symbol \(a \in S^q _0 ( {\mathbb {R}}^n ) \) is asymptotically homogeneous at infinity if \(\partial _{\xi } a \in S^{q - \varepsilon } _0 ({\mathbb {R}}^n ) \) for some \(\varepsilon > 0 \) and if
exists for all \(\xi \in {\mathbb {R}}^n \setminus 0 . \)
Note that \(a_{\infty } = a \) if a is homogeneous of order q. We will also assume for simplicity that \(\lim _{\eta \rightarrow 0 } |\eta |^{\kappa } \widehat{\varphi } (\eta ) | =: A(\varphi ) =: A \) exists: if not, one has to replace A by the corresponding liminf or limsup, and replace equalities by inequalities at the appropriate places in Theorem 6.8 below. Recall that \(A > 0 \) by the definition of the Buhmann class.
Lemma 6.7
If \(a \in S_0 ^q ({\mathbb {R}}^n ) \), \(q > 0 \), is asymptotically homogeneous at infinity, then
exists and is equal to \(g_{a, \kappa } = A^{-1 } \sum _{k \ne 0 } a_{\infty } (2 \pi k ) \, \widehat{\varphi } (2 \pi k ) . \)
Proof
The hypotheses on a easily imply that
for all \(k \ne 0 \) and all \(\xi . \) Since \(q > 0 \) it follows that for each \(\xi \in {\mathbb {R}}^n \),
where the interchange of summation and limit can be justified by dominated convergence, using that \(N > n + q . \) Since
the lemma follows. \(\square \)
Theorem 6.8
Suppose that \(a \in S^q _0 ({\mathbb {R}}^n ) \) is a symbol of order \(q > 0 \) with non-negative real part which is asymptotically homogeneous at infinity, and that \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) with order of singularity \(\kappa \ge q \) and decay rate \(N > n + q . \) Let the initial value f be as in theorem 6.4, so that in particular, \(\widehat{f }|_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{\kappa } ({\mathbb {R}}^n ) . \) Then if \(\kappa > q \),
while if \(\kappa = q \) this limit equals
where we note that \(\mathrm{Re } \, g_{a, \kappa } \ge 0 . \)
Proof
First of all, since \(\mathrm{Re } \, G_a ^* \ge 0 \) and \(|| \widehat{s_h [f ] } - \widehat{f } ||_1 = O(h^{\kappa } ) \), the limit we have to compute is equal to
If \(\kappa > q \) then for each \(\xi \), \(G_a ^* (\xi , h ) - a(\xi ) \rightarrow 0 \) as \(h \rightarrow 0 \), and consequently, using the Taylor expansion of the exponential function,
while if \(\kappa = q \), this limit equals \(| e^{ - t g_{a , \kappa } |\xi |^{\kappa } } - 1 | . \) The dominated convergence theorem and estimate (71) then show that
exists and is equal to the right hand side of (79) if \(r > 0 \) is sufficiently small. The integral over \(h |\xi | \ge r \) can as before be bounded by \(C h^{\kappa } || \widehat{f } ||_{1 , \kappa } \) which goes to 0 when multiplied by \(h^{ - (\kappa - q ) } \) since \(q > 0 \), which proves the theorem. \(\square \)
The theorem shows that the approximation order \(O(h^{\kappa - q } ) \) is exact if \(g_{a , \kappa } \ne 0 . \) The latter can be 0, for example if \(a_{\infty } (\eta ) \) is odd and \(\widehat{\varphi } (\eta ) \) is even, but then \(a_{\infty } \) has to be purely imaginary, to comply with (64)). A concrete example is given by the constant-coefficient transport equation \(\partial _t u +v \cdot \nabla u = 0 \), \(v \in {\mathbb {R}}^n . \)
Remark 6.9
We include some observations on the case of \(q \le 0 \), which can be relevant for applications. For example, the transition probability densities of a, possibly multi-dimensional, compound Poisson process satisfy a pseudo-differential evolution equation of the of the form (62): if the process has intensity \(\lambda > 0 \), and if \(c(\xi ) \) is the characteristic function of the (identically distributed and independent) jumps, the symbol equals \(a(\xi ) = \lambda (1 - c(\xi ) ) \) which has non-negative real part since \(|c(\xi ) | \le 1 . \) If the jumps have moments of all order then a is in \(S^0 _0 ({\mathbb {R}}^n ) \) (as noted before, finitely many derivatives, and thus finitely many moments, would suffice). One can use the scheme to evaluate the expectation of functions of the process, taking the function in question as initial value: see Remark 6.11(i) below.
If \(a \in S^q _0 \) with \(q = 0 \), we have to subtract \(\frac{1 }{2 } l_{\kappa } (\varphi ) a(\xi ) \) from the right hand side of (78) (where we recall that \(l_{\kappa } (\varphi ) = 2 A^{-1 } \sum _{k \ne 0 } \widehat{\varphi } (2 \pi k ) \)), while if \(q < 0 \),
where a no longer needs to be asymptotically homogeneous. Note the \(h^{\kappa } \) in the denominator: the limit would diverge with a \(h^{\kappa - q } \) if \(q < 0 . \) When analyzing the limit (79) for an \(a \in S^0 _0 \) we can no longer discard the contribution of \(e^{- t G_a ^* } ( \widehat{s_h [f ] } - \widehat{f } ) \), but one can show that if \(\widehat{f } \) has for example compact support, and if \(a \in S^0 _0 \) is asymptotically homogeneous, then
where it is no longer needed that the real part of a be positive (since a is bounded). If \(a \in S_0 ^q \) with \(q < 0 \), this result remains true with \(g_{a , \kappa } \) replaced by 0. In particular, the best order of approximation is \(h^{\kappa } \) in all these cases. We skip the details.
6.4 Approximate Approximation 2
We cannot deduce from Lemma 6.7 a precise approximate approximation estimate in terms of \(g_{a , \kappa } \), similar to Theorem 5.8, since the limit is not "uniform in \(h |\xi | \)". In fact, lower order parts of the symbol will necessarily occur in such an estimate. We will for simplicity limit the discussion to constant coefficient partial differential operators, and therefore take \(a(\xi ) = p(\xi ) \), a polynomial of degree \(q \in {\mathbb {N}} \) with non-negative real part: \(\mathrm{Re } \, p(\xi ) \ge 0 . \) Let
with \(p_{q - j } (\xi ) \) homogeneous of degree \(q - j . \) We then can write \(G_{p_{\nu } } ^* (\xi ; h ) = h^{-\nu } G_{p_{\nu } } (h\xi ) \), where
and we have
for \(\nu > 0 \), while \(G_{p_0 } = p_0 \) since \(p_0 \) is a constant. Since
it follows that there exists for each \(\varepsilon > 0 \) a \(\rho = \rho (\varepsilon ) > 0 \) such that
Since \(\mathrm{Re } \left( p (\xi ) - G_p ^* (\xi ; h ) \right) \le |h \xi |^{\kappa } \mathrm{Re } \, p (\xi ) < \varepsilon \mathrm{Re } \, p(\xi ) \) if \(h |\xi | \) is sufficiently small, the arguments used to prove Theorem 6.4 will now give the following estimate:
Theorem 6.10
For all \(\varepsilon > 0 \) there exists a \(C_{\varepsilon } > 0 \) such that if \(s > \kappa - q \) and if f has polynomial growth of order strictly less than \(\kappa \) such that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 _{\kappa } ({\mathbb {R}}^n ) \cap L^1 _s ({\mathbb {R}}^n ) \),
Here \( | g_{p_{q - j } , \kappa } | + \varepsilon \) can be replaced by \((1 + \varepsilon ) | g_{p_{q - j } , \kappa } | \) if \(g_{p_{q - j } , \kappa } \ne 0 . \)
6.5 Homogeneous Symbols
We finally indicate what changes to make when allowing homogeneous symbols such as \(|\xi |^q \) or, more generally, symbols in the class
First of all, \(a(D) L_1 (x) \) will now only decay as \(|x |^{- q - n } \) at infinity: see Remark A.3. This implies that for the scheme to be well-defined (as a coupled system of differential equations on \(h{\mathbb {Z}}^n \)) we have to restrict ourselves to initial values of polynomial growth of order strictly less than q: \(f \in L^{\infty } _{- (q - \varepsilon ) } ({\mathbb {R}}^n ) . \) Next, a and \(e^{- t a } \) will now be in the Hölder space \(C^{\lceil q \rceil - 1 , \lambda } \) with \(\lambda = q - (\lceil q \rceil - 1 ) \), as will \(G_a ^* \) and \(e^{-t G_a ^* } \), assuming that \(q \le \kappa . \) Using the arguments of Appendix B, one shows that if f has polynomial growth of order strictly less than q and if its Fourier transform satisfies
then \(e^{- t a } \widehat{f } \), \(e^{- t G_a ^* } \widehat{f } \) are all well-defined as tempered distributions (assuming of course that \(\mathrm{Re } \, a \ge 0 \)). The same is true for and \(e^{- t G_a ^* } \widehat{s_h [f ] } \) since \(\widehat{s_h [f ] } - \widehat{f } \) is an \(L^1 \)-function: cf. (104), the conditions for this formula being met since \(q \le \kappa . \) One checks that this is again the Fourier transform of \(u_h (\cdot , t ) \) and the basic identity (69) for the Fourier transform of \(u (\cdot , t ) - u_h (\cdot , t ) \) will still hold, as will the basic estimate (68) of proposition 6.3 (which only uses the bound for a, not the ones for its derivatives). We then see that our convergence theorem 6.4 remains valid provided we restrict to initial values f which have polynomial growth of order \(< q \) and satisfiy (83). Combining this with the condition that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \) belong to \(L^1 _{\kappa } ({\mathbb {R}}^n ) \) and remembering that \(q \le \kappa \), this means that the RBF scheme for a symbol \(a \in \mathring{S }^q _1 ({\mathbb {R}} \setminus 0 ) \) is convergent of order \(\kappa - q \) if the initial value \(f \in L^{\infty } _{ - q + \varepsilon } \) for some \(\varepsilon > 0 \) and if \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \in L^1 ( {\mathbb {R}}^n , \max (|\xi |^q, |\xi |^{\kappa } ) ) . \) The same observation applies to the approximate approximation estimates and to Theorem 6.8 on the exact convergence rate.
We end this section with some examples of pseudo-differential evolution equations to which our results apply.
Examples 6.11
(i) Kolmogorov equations of Lévy processes Let \((X_t )_{t \ge 0 } \) be a (multi-dimensional) Lévy process on \({\mathbb {R}}^n . \) Recall that according to the Lévy - Khintchine theorem such a process is completely characterized by its characteristic function \({\mathbb {E}} \left( e^{ i (\xi , X_t ) } \right) \) which is of the form \(e^{t \psi (\xi ) } \) where
with \(\mu \in {\mathbb {R}}^n \), \(\Sigma \) a positive semi-definite linear operator on \({\mathbb {R}}^n \), \(\nu \) a positive Borel measure on \({\mathbb {R}}^n \setminus 0 \) such that
and \(\chi \) a compactly supported bounded function which is equal to 1 on a neighborhood of 0, and which can be taken smooth, if necessary. The measure \(\nu \) is called the Lévy measure of the process.
If, for a given f, we let \(u(x, t ) = {\mathbb {E}} (f (x + X_t ) ) \), then u satisfies (62) with \(a(\xi ) := - \psi (\xi ) \) and initial value f (recall that \(X_0 = 0 \) a.s., by definition). Note that \(a(\xi ) \) satisfies (64) since
Under appropriate hypotheses on the Lévy-measure \(\nu \) one can derive symbol-type estimates for \(a(\xi ) . \) For example, if \(d\nu (x ) = |x |^{- Y } h(x) dx \) with \(Y < n + 2 \), and if h(x) is a rapidly decreasing continuous function, then \(a \in S^2 _0 \) if \(\Sigma \ne 0 \), and \(a \in S^{\max (Y - n , \, 0 ) } _0 \) if \(\Sigma = 0 \) and \(Y \notin {\mathbb {N}} \) or \(Y < n \): see Remark A.5 in Appendix A. (If \(Y = n \) or \(n + 1 \) there is an additional logarithmic factor.) Processes of this type are used in mathematical finance and the associated evolution equations were solved numerically in [3, 9, 10] using the semi-discrete RBF scheme of the present paper, with different choices of basis functions. If \(\Sigma = 0 \) and if \(\nu \) is a finite measure on \({\mathbb {R}}^n \) of total mass \(\lambda \), we have the special case of a (multi-dimensional) compound Poisson process with intensity \(\lambda \) and drift \(\mu + \int _{{\mathbb {R}}^n } x \chi d\nu (x) \), \(\lambda ^{-1 } \nu \) being the probability law of the jumps. The symbol will be in \(S^0 _0 \) if \(\nu \) has moments of all order.
One might consider using the RBF-scheme to compute the transition probability densities of such Lévy processes, as an alternative to inverting the characteristic function of the process. This was done in Rad et al. [28] for Brownian motion-driven diffusion processes, where the problem of how to implement an initial Dirac delta distribution in an RBF scheme was solved by first using a small-time approximation for the transition probability due to Aït-Sahalia.
The RBF-scheme was used in [3] to compute put option prices in the so-called CGMY model [8] (see also [12]), in which log-asset prices follow a one-dimensional Lévy process (\(n = 1 \)) with Lévy measure \(|x |^{- Y } h(x) dx \) with \(Y < 3 \) and h a negative exponential on the positive and negative axis. (Such processes are called truncated Lévy-flights in the physics literature.) This problem attracted considerable attention in the computational finance community, where it was principally treated using finite difference schemes: see [3] for further discussion and references. Finite difference schemes require discretisation of the operator a(D) , which leads to technical difficulties if the Lévy measure is not integrable at 0 (\(Y > 1 \)) and in particular if \(Y > 2 \), when the process is of infinite variation. For such processes, finite difference schemes perform less well, and the numerical performance of the RBF scheme was found to be superior in terms of empirical convergence rates. We briefly comment on the results of [3] in the light of Theorems 6.4 and 6.5 .
Assuming for simplicity that \(Y \ne 1 , 2 \), the symbol of the generator of the CGMY process is in \(S^{\max (Y - 1 , 0 ) } _0 \) if \(\Sigma = 0 \) and in \(S^2 _0 \) if the process has a Brownian motion component, \(\Sigma \ne 0 . \) The RBF \(\varphi \) used in [3] was Hardy’s multi-quadric (61) in dimension 1 with a scale parameter of \(c = 4 . \) Since the multi-quadric belongs to \(\mathfrak {B }_{2, N } ({\mathbb {R}} ) \) for all \(N > 1 \), Theorem 6.4 predicts a convergence rate of respectively \(2 - \max (Y - 1 , 0 ) \) if \(\Sigma = 0 \), and 0 if \(\Sigma \ne 0 \). The numerical results of [3] suggested however that, over the range of h’s used, the scheme converges at a rate of \(h^2 \) for all of the CGMY processes which were considered, including when Y is close to 3 or when the process has a Brownian motion component: \(\Sigma \ne 0 . \) This can be explained by approximate approximation: from the asymptotics of the Bessel function, we find that \(\widehat{\varphi } (\eta ) \simeq \sqrt{2 \pi c } |\eta |^{- 3/2 } e^{- 3/2 } e^{- c |\eta | } \), and therefore, to good approximation, \(l_{\kappa } (\varphi ) \simeq \sqrt{c } / (2 \pi ) e^{- 2 \pi c } \), which is of the order of \(10^{-11 } \) while for example \(g_{\kappa } (\varphi ) \simeq 2 \pi \sqrt{c } e^{- 2 \pi c } \) is of the order of \(10^{-9 } \), with a similar estimate for \(\overline{l }_{\kappa , q } \), \(q = \max (Y - 1, 0 ) \), cf. (75). The range of h’s considered was [0.005, 0.2] so that \(g_{\kappa } (\varphi ) h^0 \) remained smaller than \(h^2 \) by an order of at least three.
(ii) The special case of symmetric stable Lévy-processes leads to the fractional heat equation:
with \(s \in (0, 1 ) \), which also has applications in physics (anomalous diffusion, soft matter, turbulence: see for example [21] and its references). Here the symbol, \(a(\xi ) = |\xi |^{2s } \in \mathring{S }^{2s } _1 ({\mathbb {R}}^n \setminus 0 ) \) and we have to restrict the initial values f, as explained in the preceding subsection.
(iii) Non-parabolic equations, such as the transport equation \(\partial _t u = v \cdot \nabla u \), the free Schrödinger equation \(i \partial _t u = - \Delta u \) or the half-wave equations \(\partial _t u = \pm i \sqrt{ - \Delta } u \): for these, the symbol \(a(\xi ) \) is purely imaginary. For the classical wave equation, \(\Box u = \partial _t ^2 u - \Delta u = 0 \), one can use factorisation, \(\Box u = - (i \partial _t + \sqrt{ - \Delta } ) (i \partial _t + \sqrt{- \Delta } ) u \) or, alternatively, rewrite the equation as a first order \(2 \times 2 \)-system and extend the convergence analysis of this paper to matrix-valued symbols \(a(\xi ) . \)
Notes
The index \(\kappa \) is a reminder of the degree of the singularity of \(\widehat{\varphi } \) at 0, and therefore of the convergence rate of the RBF interpolants.
In the case of the heat equation (\(a(\xi ) = |\xi |^2 \)), \(a - G_a ^* \) was negative for small enough \(h |\xi | \), but this is not clear for \(\mathrm{Re } (a - G_a ^* ) \) for general a, in particular when \(\mathrm{Re } \, a \) is not elliptic.
Note that \(\lfloor p \rfloor + n + 1 \ge 1 \) since \(p > - n . \)
e.g. by using the Taylor formula with integral remainder in the form
$$\begin{aligned} \psi (\xi ) - \sum _{|\alpha | \le \kappa - 1 } \psi ^{(\alpha ) } (0) \frac{\xi ^{\alpha } }{\alpha ! } = \int \limits _0 ^1 \frac{((1 - s )^{\kappa - 1 } }{(\kappa - 1 ) ! } \frac{d ^{\kappa } }{d s^{\kappa } } \psi _{\xi } (s) ds , \end{aligned}$$where \(\psi _{\xi } (s) := \psi (s \xi ) \)
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Appendices
Appendix
Appendix A: Proof of Theorem 2.3
We prove the existence and main properties of the cardinal function associated to a basis function \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) as stated in Theorem 2.3. As mentioned, this was already done by Buhmann [4, 5] for a more restricted class of radial basis functions. The main difference in our treatment and that of [4] is the use of Lemma A.2 below, relating the decay at infinity of the Fourier transform of a function with its behavior in 0, which allows one to go beyond the class of basis functions considered by Buhmann.
It may be interesting to observe that the estimates (4) are similar to conditions (63) on symbols of pseudodifferential operators, except that the latter restrict the behavior at infinity instead of that at 0. Indeed, if (4) were required for all orders \(\alpha \) (with constants depending on \(\alpha \)), then \((1 - \chi (\xi ) ) \widehat{\varphi } (\xi /|\xi |^2 ) \in S^{\kappa } _1 ({\mathbb {R}}^n ) \) if \(\chi \in C_c ^{\infty }({\mathbb {R}}^n ) \) is equal to 1 on a neighborhood of 0. From this point of view, (5) corresponds to having an elliptic symbol, whence our terminology.
As a preliminary remark, we note that conditions (ii) and (iii) of Definition 2.1 imply that
as an easy proof by induction shows.
Turning to the proof of theorem 2.3, following Buhmann [4, 5], we start by defining \(L_1 \) as the inverse Fourier transform of the right hand side of (9), observing that since, by condition (iv) of Definition 2.1, the latter is an integrable function, \(L_1 \) is a well-defined continuous function. We first show that \(L_1 (x) \) has the proper decay at infinity.
Theorem A.1
Let \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \), and let
Then there exists a positive constant C such that
The proof will use the following lemma, which basically is a special case of a classical estimate for kernels of convolution operators: see Stein [35], Proposition 2 of Chapter VI, Sect. 4.4.
Lemma A.2
Let \(p > - n \) and let \(a \in C^{\lfloor p \rfloor + n + 1 } ({\mathbb {R}}^n \setminus 0 ) \) be supported in some ball B(0, R) such thatFootnote 3
Then the inverse Fourier transform \(k = \mathcal {F }^{-1 } (a) \) satisfies
with a constant \(C_1 \le c_n C \), where \(c_n \) only depends on n.
Stein in fact shows that if (88) is satisfied at all orders, without a necessarily being compactly supported, then k can be identified with a \(C^{\infty } \)-function away from 0, satisfying \(|\partial _x ^{\alpha } k (x) | \le C_{\alpha } |x |^{ - p - n - |\alpha | } \) for all \(\alpha \) and all x. This result is stated and proven there for \(p = 0 \), but the proof generalizes to any \(p > -n . \) We only need this estimate for k(x) itself, in which case we only need (88) for the limited number of derivatives of a indicated, and we furthermore only need it for large |x| (note that if a has compact support, k is continuous, even \(C^{\infty } \), and Stein‘s estimate for k(x) at 0 becomes trivial). The proof in [35] uses the Paley–Littlewood decomposition. An elementary prove of Lemma A.2 can be given by writing
where \(\chi \in C^{\infty }({\mathbb {R}}^n ) \) with bounded derivatives such that \(\chi (\xi ) = 0 \) for \(|\xi | \le 1 \), \(\chi (\xi ) = 1 \) for \(|\xi | \ge 2 \), and integrating the first integral by parts \(\lfloor p \rfloor + n + 1 \) times.
Proof of theorem A.1
Let \(\chi _0 \in C^{\infty } _c ({\mathbb {R}}^n ) \) such that \(\chi _0 (\eta ) = 1 \) in a neighbourhood of 0 and \(\mathrm{supp }(\chi _0 ) \subset (- \pi , \pi ) ^k . \) For \(k \in {\mathbb {Z}}^n \), define \(\chi _k \) by \(\chi _k (\eta ) := \chi _0(\eta + 2 \pi k ) \) and note that the supports of the \(\chi _k \) are disjoint. Finally, let \(\chi _c := 1 - \sum _k \chi _k \) ("c" for "complement"), so that \(\chi _c \) together with the \(\chi _k \)’s form a partition of unit. Then
where
We examine the decay in x of each term separately.
Decay of \(\ell _c \) The function \(\chi _c (\eta ) / \sum _k \widehat{\varphi } (\eta + 2 \pi k ) \) is in \(C_b ^{\lfloor \kappa \rfloor + n + 1 } ({\mathbb {R}}^n ) \), since the denominator is a strictly positive periodic function which is \(C_b ^{\lfloor \kappa \rfloor + n + 1 } \) on the complement of \((2 \pi {\mathbb {Z}} )^n \) and therefore on the support of \(\chi _c . \) Multiplying with \(\widehat{\varphi } \), we find that \(\widehat{\ell }_c (\eta ) \) is \(C^{\lfloor \kappa \rfloor + n + 1 } \) with integrable derivatives of all orders, which implies by the usual integration by parts argument that \(|\ell _c (x) | \le C (1 + |x| )^{-(\lfloor \kappa \rfloor + n + 1 ) } \le C (1 + |x| )^{- \kappa - n } . \)
Note that \(\widehat{L }_1 (\eta ) \) is at best \(C^{\lfloor \kappa \rfloor } \) in the points of \(2 \pi {\mathbb {Z}}^n \), so integration by parts will not give the required decay for of \(\ell _k \), \(k \in {\mathbb {Z}}^n . \) We use Lemma A.2 instead.
Decay of \(\ell _0 \) Since
and since \(\sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k ) \) is \(C^{\lfloor \kappa \rfloor + n + 1 } \) on the suport of \(\chi _0 \), the estimates (85) easily imply that \(\widehat{\ell }_0 (\eta ) - \chi _0 (\eta ) \) satisfies condition (88) of Lemma A.2 with \(p = \kappa \) (for \(\psi := \widehat{\varphi } ^{-1 } \sum _{k \ne 0 } \widehat{\varphi } (\cdot + 2 \pi k ) \) does, and then also \(\psi / (1 + \psi ) \)). It follows that \(|\ell _0 (x) | \le C (1 + |x |)^{- \kappa - n } \), since \(\mathcal {F }^{-1 } (\chi _0 ) \) is rapidly decreasing.
Decay of \(\ell _k \), \(k \ne 0, c \) This is similar, except that we have to pay attention to the size of the constant in front of the \((1 + |x| )^{ - \kappa - n } . \) The Fourier transform \(\widehat{\ell }_k (\eta ) \) will now be supported near \(\eta = - 2 \pi k . \) Shifting by \(2 \pi k \), we see that
is supported in a small neighbourhood of 0, with derivatives of order \(|\alpha | \le \lfloor \kappa \rfloor + n + 1 \) bounded by \(C (1 + |k | )^{-N } |\eta |^{\kappa - |\alpha | } \), with C independent of k. Lemma A.2 then implies that
Since \(N > n \), summation over \(k \in {\mathbb {Z}}^n \) completes the proof. \(\square \)
Remark A.3
The same arguments prove Theorem 6.1: if \(a \in S^q _0 ({\mathbb {R}}^n ) \) we have \(\widehat{a(D)(L_1)} = a \widehat{\ell }_c + \sum _k a \widehat{\ell _k } \), and \(a \widehat{\ell }_c \) has \(\lceil \kappa \rceil + n + 1 \) derivatives which are integrable since \(N - q > n \), as has \(a \chi _0 . \) Furthermore, \(a \chi _0 (\widehat{L }_1 - 1 ) \) and \(a \chi _k \widehat{L }_1 \), \(k \ne 0 \) satisfy the hypothesis (88) of Lemma A.2, the latter with constants bounded by \(C |k |^{-(N - q ) } \) which are summable since \(N > n + q . \) If \(a(\xi ) = \xi ^{\alpha } \) we obtain theorem 2.7.
Finally, for symbols a in \(\mathring{S }^q _1 ({\mathbb {R}}^n \setminus 0 ) \) the estimate for \(\mathcal {F }^{-1 } (\chi _0 a \widehat{L }_1 ) = \mathcal {F }^{-1 } ( a \chi _0 ) + \mathcal {F }^{-1 } ( a \chi _0 (\widehat{L }_1 - 1 ) ) \) changes: Lemma A.2 shows that the first term decays like \((1 + |x | )^{- q - n } \) while the second one decays as \((1 + |x | )^{- q - \kappa - n } . \) The other terms in the decomposition of \(\mathcal {F }^{-1 } (a \widehat{L }_1 ) \) can be estimated by a constant times \((1 + |x )^{- \kappa - n } \), as before.
We continue with the proof of theorem 2.3. The expressions for \(\widehat{\ell }_k \) above also show that \(\widehat{L }_1 \) satisfies the Strang-Fix conditions (10). Once we have defined \(L_1 \) through its Fourier transform, it is immediate to check that \(L_1 (k) = \delta _{0k } \) for \(k \in {\mathbb {Z}}^n \): indeed, by the \(2 \pi \)-periodicity of the denominator, writing the integral over \({\mathbb {R}}^n \) as a sum of integrals over translates of \((-\pi , \pi )^n \),
It remains to recognise \(L_1 \) as a sum of translates of \(\varphi . \) To show this we first write the denominator of (9) as a Fourier series:
One verifies by the similar arguments as the ones of the proof of theorem A.1 that
write
and estimate the first integral using Lemma A.2 and the second by integrating by parts.
It follows that (92) converges absolutely. We then claim that
where the series converges absolutely and uniformly on compacta, by (93), since \(\varphi (x) \) grows at most as \((1 + |x \ )^{\kappa - \varepsilon } \), by assumption. Formally, (94) follows by writing
except that the final step does not make sense for an arbitrary \(\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n ) \) since \(\widehat{\varphi } (\eta ) \) will not be integrable in 0 if \(\kappa > n . \) We have to carefully distinguish between the tempered distribution \(\widehat{\varphi } \) and the locally integrable function \(\eta \rightarrow \widehat{\varphi } (\eta ) \) with which it can be identified on \({\mathbb {R}}^n \setminus 0 . \) The relation between the two is given by the following identity: there exist constants \(c_{\alpha } \), \(|\alpha | \le \lceil \kappa \rceil - 1 \) such that for all \(\psi \in \mathcal {S }({\mathbb {R}}^n ) \),
where the first sum is empty if \(\kappa < n . \) Indeed, the first integral converges since \(|\eta |^{\lfloor \kappa \rfloor - n + 1 } \widehat{\varphi } (\eta ) \) has an integrable singularity at 0. The sum of the two integrals on the right defines a tempered distribution. If we denote this distribution by \(\Lambda _{\widehat{\varphi } } \) then the restriction of \(\Lambda _{\widehat{\varphi } } \) to \({\mathbb {R}}^n \setminus 0 \) can be identified with the function \(\widehat{\varphi } (\eta ) . \) The difference \(\widehat{\varphi } - \Lambda _{\widehat{\varphi } } \) is then supported in 0, and therefore a linear combination \(\sum _{|\alpha | \le p } c_{\alpha } \delta _0 ^{(\alpha ) } \) of derivatives of the delta distribution in 0. To bound p, we use the following lemma, whose proof we postpone till the end of his section:
Lemma A.4
The inverse Fourier transform \(\mathcal {F }^{-1 } \left( \Lambda _{\widehat{\varphi } } \right) \) is a continuous function which is bounded if \(\kappa < n \) while if \(\kappa \ge n \) it is bounded by \(C (|x |^{\kappa - n } +1 ) \) if \(\kappa \notin {\mathbb {N}} \) and by \(C (|x |^{\kappa - n } \log |x | + 1 ) \) if \(\kappa - n \in {\mathbb {N}} . \)
Since the inverse Fourier transform of \(\sum _{|\alpha | \le p } c_{\alpha } \delta _0 ^{(\alpha ) } \) is a polynomial of order p, and since, by assumption, \(\varphi (x) \) has polynomial growth of order strictly less than \(\kappa \), it follows that \(p < \kappa \), which is equivalent to \(p \le \lceil \kappa \rceil - 1 . \)
We now use (95) to prove that (94) holds as tempered distributions, that is, if \(\psi \in \mathcal {S }({\mathbb {R}}^n ) \), then
If we let
then \(\Psi \) is \(C^{\lfloor \kappa \rfloor } \) if \(\kappa \notin {\mathbb {N}} \), and \(C^{\kappa - 1 , 1 } \) if \(\kappa \in {\mathbb {N}}^* \), with all its derivatives rapidly decreasing. To obtain a function in the Schwartz class \(\mathcal {S }({\mathbb {R}}^n ) \) we convolve with \(\chi _{\varepsilon } (x) := \varepsilon ^{-n } \chi (x/\varepsilon ) \), where \(\chi \in C^{\infty }_c ({\mathbb {R}}^n ) \) with integral 1. Let \(\Psi _{\varepsilon } := \chi _{\varepsilon } * \Psi . \) Then we first claim that
To show this it suffices to consider the case that \(\kappa \ge n \), since \(\widehat{\varphi } \) is integrable if \(\kappa < n \) and \(\Psi _{\varepsilon } \) converges uniformly. If we assume for example that \(\mathrm{supp }(\chi ) \subset B(0, 1 ) \) then by the Taylor expansion with remainder there exists a constant \(C > 0 \) such that for all \(\varepsilon \le 1 \),
where we note that if \(n \ge 2 \) or if \(\kappa \notin {\mathbb {N}}^* \), then derivatives of \(\Psi \) of order \(\lfloor \kappa \rfloor - n + 1 \) exist, while if \(n = 1 \) and \(\kappa \in {\mathbb {N}}^* \), these derivatives exist a.e. but are uniformly bounded, and the estimate remains true. Next, \(\Psi ^{(\alpha ) } _{\varepsilon } (x) \rightarrow \Psi ^{(\alpha ) } (x) \) for \(|\alpha | \le \lceil \kappa \rceil - 1 \) since \(\Psi \) is \( C^{\lceil \kappa \rceil - 1 } . \) Furthermore, since \((\sum _k \widehat{\varphi } (\eta + 2 \pi k ) )^{-1 } \) vanishes of order \(\kappa \) in 0, it follows that \(\Psi ^{(\alpha ) } (0) = 0 \) for \(|\alpha | \le \lceil \kappa \rceil - 1 \) and (97) follows by dominated convergence.
Since \(\Psi _{\varepsilon } \) is Schwartz-class, we have \(\langle \widehat{\varphi } , \Psi _{\varepsilon } \rangle = \langle \varphi , \widehat{\Psi }_{\varepsilon } \rangle . \) By (92) \(\Psi = \left( \sum _k c_k e^{-i (k, \eta ) } \right) \psi (\eta ) \), which can be interpreted as the product of a tempered distribution and a test function, whose Fourier transform equals
One easily verifies using (93) that \(|\widehat{\Psi } (x) | \le C (1 + |x| )^{- \kappa - n } . \)
Since \(\widehat{\Psi }_{\varepsilon } (x) = \widehat{\chi } (\varepsilon x ) \widehat{\Psi } (x) \), and since \(|\varphi (x) | \ \le C (1 + |x | )^{\kappa - \rho } \) for some \(\rho > 0 \), Lebesgue’s dominated convergence theorem then shows that
Finally, one checks that the functions \((x, k ) \rightarrow c_k \varphi (x) \widehat{\psi } (x - k ) \) and \((x, k ) \rightarrow c_k \varphi (x + k ) \widehat{\psi } (x) \) are integrable on \({\mathbb {R}}^n \times {\mathbb {Z}}^n \) with respect to the product of the Lebesgue measure and the counting measure. A double application of Fubini’s theorem then shows that the right hand side equals
which proves (96). The pointwise identity (94) follows since both sides are continuous functions.
Proof of Lemma A.4
We give a proof for convenience of the reader. If \(\kappa < n \), \(\widehat{\varphi } \) is integrable, and its inverse Fourier transform is a bounded continuous function, so suppose that \(\kappa \ge n . \) Since \(\mathbf {1 }_{\{ |\eta | \ge 1 \} } \widehat{\varphi } (\eta ) \) is integrable, its inverse Fourier transform is a bounded continuous function, and it therefore suffices to examine the inverse Fourier transform of the tempered distribution defined by the first integral on the right hand side of (95). This distribution being of compact support, its inverse Fourier transform is the function k(x) obtained by taking \(\psi (\eta ) = (2\pi )^{-n } e^{i(x, \eta ) } \):
where \(\nu := \lfloor \kappa \rfloor - n . \) This is a \(C^{\infty } \)-function which can be bounded by
Split the integral into an integral over \(|\eta | \le c \) and one over the complement, where \(c > 0 \) is some fixed number and where we assume wlog that \(|x | > c . \) The first integral converges absolutely, since
and \(\nu - \kappa + n = \lfloor \kappa \rfloor - \kappa > -1 \), and we can bound its contribution to k(x) by \(C |x |^{\kappa - n } . \) If \(\kappa \notin {\mathbb {N}} \) the integral over \(| \eta | > c \) can be bounded by a constant times
which is bounded by \(C (|x |^{\kappa - n} + 1 ) \) since \(\nu - \kappa + n < 0\). Finally, if \(n \le \kappa \in {\mathbb {N}} \), then \(\nu = \kappa - n \) and
\(\square \)
Remark A.5
More generally, if F is a measurable function such that \(F(\eta ) = O(|\eta |^{- \kappa } ) \) near 0 and \(|\eta |^r F \cdot \mathbf {1 }_{ \{ |\eta | \ge 1 \} } \in L^1 ({\mathbb {R}}^n ) \) for some \(r \in {\mathbb {N}} \), then \(K := \mathcal {F }^{-1 } (\Lambda _F ) \), with \(\Lambda _F \) defined by the first two terms of (95) with \(\widehat{\varphi } \) replaced by F, is a \(C^r \)-function whose derivatives can be bounded by
where \(|\alpha | \le r . \) This immediately implies symbol estimates for the generators of pure-jump Lévy processes (\(\Sigma = 0 \)) with Lévy measure of the form
where \(h = h(\eta ) \) is a rapidly decreasing function. Taking \(\chi \) equal to the characteristic function of the unit ball in the Lévy–Khintchine formula (84), and setting \(\mu = 0 \) (without essential loss of generality), we see that the \(\psi (\xi ) \) is exactly the inverse Fourier transform of the distribution \(\Lambda _{ |\eta |^{- Y } h(\eta ) } \), modulo a constant which equals \(\int (1 - \chi ) d\nu (x) . \) This certainly implies that \(\psi \in S^{\max (Y - n , \, 0 ) } _0 \) if \(Y \notin {\mathbb {N}} \) (there is an additional decay-factor of \(|\xi |^{-1 } \) for the first derivative if \(n< Y < n + 1 \)), while \(\psi / \log (|\xi |^2 + 1 ) \in S_0 ^{Y - n } \) if \(Y = n \) or \(Y = n + 1 . \)
Appendix B: Some Technical Proofs
1.1 Appendix B.1: Proof of Lemma 3.5
Let \(F \in L^1 \left( {\mathbb {R}}^n \setminus 0 , (|\xi |^{\kappa } \wedge 1 ) d\xi \right) \), with \(\kappa \ge 0 . \) Then F gives rise to a tempered distribution \(\Lambda _F \in \mathcal {S }' ({\mathbb {R}}^n ) \) defined as follows: if g is a bounded compactly supported function which is equal to 1 on a neighbourhood of 0, then
cf. (95). The integral converges since \(\psi - \sum _{|\alpha | \le \lceil \kappa \rceil - 1 } \psi ^{(\alpha ) } (0) \xi ^{\alpha } / \alpha ! = O(|\xi |^{\lceil \kappa \rceil } ) = O(|\xi |^{\kappa } ) \) in a neighbourhood of 0 and defines a distribution of order \(\lceil \kappa \rceil - 1 . \) Note that \(\Lambda _F \) coincides on \({\mathbb {R}}^n \setminus 0 \) with the locally integrable function F.
We next observe that \(\Lambda _F \) extends to a continuous linear functional on the Hölder space \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } := C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) \) with \(\lambda = \kappa - (\lceil \kappa \rceil - 1 ) . \) Indeed, if \(\psi \in C^{K , \lambda } ({\mathbb {R}}^n ) \), then the Taylor expansion formula with integral remainder term easily implies that
which shows, with \(K = \lceil \kappa \rceil - 1 \) and \(\lambda = \kappa - (\lceil \kappa \rceil - 1 ) \), that \(\langle \Lambda _F , \psi \rangle \) is well-defined and continuous.
We can, in particular, let \(\Lambda _F \) act on the imaginary exponentials \(\xi \rightarrow e^{i (x, \xi ) } . \) The function
is then found to be bounded by \( C (1 + |x | )^{\kappa } \), since \(|| e^{i(x, \xi ) } ||_{K , \lambda } \le C (1 + |x |^{K + \lambda } ) \), and one easily verifies that the inverse Fourier transform of \(\Lambda _F \) coincides with \(\check{F } . \) If \(\kappa \in {\mathbb {N}} \) one has the stronger estimate
which can be seen as follows: write \(F= \chi F + (1 - \chi ) F \) with \(\chi \) the characteristic function of a small ball around 0. Since \((1 - \chi ) F \) is integrable, its inverse Fourier transform tends to 0 at infinity, by the Riemann-Lebesgue lemma. We can therefore wlog assume that F is supported in \(\{ g = 1 \} . \) If we apply (99) with \(\psi (\xi ) = e^{i (x, \xi ) } \) thenFootnote 4
where \(\check{F }_{\alpha } (x) \) is the inverse Fourier transform of the \(L^1 \)-function \(\xi \rightarrow \xi ^{\alpha } F(\xi ) . \) By the Riemann-Lebesgue lemma, \(\check{F }_{\alpha } (sx) \rightarrow 0 \) as \(x \rightarrow \infty \), for all \(s \in (0, 1 ] \), and the same is true for the integral over \(s \in [0, 1 ] \), by the dominated convergence theorem (the \(\check{F }_{\alpha } \) are bounded). Hence \(\check{F }(x) / |x |^{\kappa } \rightarrow 0 \) for \(x \rightarrow \infty \), as claimed.
Now let f be a measurable function on \({\mathbb {R}}^n \) of polynomial growth of order strictly less than \(\kappa \), such that its Fourier transform \(\widehat{f } \) (in the sense of tempered distributions) satisfies
We write \(\Lambda _{\widehat{f } } \) for \(\Lambda _{\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } } . \) Then \(\widehat{f } - \Lambda _{\widehat{f } } \) is a distribution which is supported in 0, and therefore of the form \(\sum _{|\alpha | \le N } c_{\alpha } \delta _0 ^{(\alpha ) } \) for certain \(N \in {\mathbb {N}} \) and \(c_{\alpha } \in {\mathbb {C}} \) with \(\sum _{|\alpha | = N } |c_{\alpha } | \ne 0 . \) Since the inverse Fourier transform of \(\widehat{f } - \Lambda _{\widehat{f } } \) is a polynomial of degree N, it follows that \(N \le \lceil \kappa \rceil - 1 \), the largest integer which is strictly smaller than \(\kappa \), since otherwise \(|f(x) - \check{F}(x)|\) would grow at a rate of at least \(|x |^{\lceil \kappa \rceil } \) in certain directions. If \(\kappa \notin {\mathbb {N}} \) this would contradict the bound \(\check{F }(x) = 0(|x |^{\kappa } ) \), and if \(\kappa \in {\mathbb {N}} \) this would contradict (101).
In follows that \(\widehat{f } = \Lambda _{\widehat{f } } \, + \, \sum _{|\alpha | \le N } c_{\alpha } \delta _0 ^{(\alpha ) } \) also extends to a continuous linear functional on \(C^{\lceil \kappa \rceil - 1 , \kappa - (\lceil \kappa \rceil - 1 ) } . \) We exploit this to define \(\Sigma _h (\widehat{f } ) \) by duality.
If \(\psi \in \mathcal {S }({\mathbb {R}}^n ) \), we let
Note that \(\Sigma _h ' \) is the formal (real) adjoint of \(\Sigma _h . \) By Lemma 2.5, \(\Sigma _h ' (\psi ) \) is \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } \) with \(\lambda = \kappa - (\lceil \kappa \rceil - 1 ) \) and uniformly bounded together with all its derivatives, since \(2 \pi h^{-1 } \)-periodic. In fact, this is true even if \(\psi \in C_b ^{\lceil \kappa \rceil - 1 , \lambda } \) with the same \(\lambda \), on account of the decay at infinity of \(\widehat{L }_1 . \) We can then define \(\Sigma _h (\widehat{f } ) \), as a tempered distribution and, more generally, as a bounded linear functional on \(C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n ) \) by
We next check that \(\Sigma _h (\widehat{f } ) \) is the Fourier transform, in distribution sense, of \(s _h [f ] . \) This is done by a standard approximation argument, with some care with the spaces in which the approximating sequence converges. We first note that we can assume without loss of generality that \(\widehat{f } \) is compactly supported: indeed, we can write \(f = f_1 + f_2 \) with \(\widehat{f }_1 \) compactly supported and \(\widehat{f }_2 \in L^1 ({\mathbb {R}}^n ) \), and we know already that \(\widehat{s _h [f_2 ] } = \Sigma _h (\widehat{f }_2 ) . \)
So let \(\widehat{f } \) be compactly supported, and let \(\chi \in C^{\infty } _c ({\mathbb {R}}^n ) \) be a non-negative symmetric function with \(\int _{{\mathbb {R}}^n } \chi d\eta = 1 . \) Let \(\chi _{\varepsilon } (\eta ) := \varepsilon ^{-n } \chi (\eta / \varepsilon ) . \) Then \(\widehat{f }* \chi _{\varepsilon } \in C^{\infty } _c ({\mathbb {R}}^n ) . \)
Lemma B.1
\(\widehat{f } * \chi _{\varepsilon } \rightarrow \widehat{f } \) in the dual of \(C^{K , \lambda } \), with \(K = \lceil \kappa \rceil - 1 \) and \(\lambda = \kappa - K . \)
Proof
On account of the symmetry of \(\chi \),
which is valid both for Schwarz-class functions \(\psi \in \mathcal {S } \) and for \(\psi \in C^{K , \lambda } . \) Write \(\psi _{\varepsilon } := \psi * \chi _{\varepsilon } . \) If \(\psi \in C^{K , \lambda } \), then \(\psi _{\varepsilon } ^{(\alpha ) } (x) \rightarrow \psi ^{(\alpha ) } (x) \) pointwise on \({\mathbb {R}}^n \) for all \(|\alpha | \le K \), while a trivial estimate shows that \(|| \psi _{\varepsilon } ^{(\alpha ) } ||_{0, \lambda } \le || \psi ^{(\alpha ) } ||_{0 , \lambda } \), uniformly in \(\varepsilon > 0 \), for \(|\alpha | = K . \) This, together with the remainder estimate (100), the integrability of \(\widehat{f } (\xi ) (|\xi |^{\kappa } \wedge 1 ) \) and Lebesgue’s dominated convergence theorem, implies that \( \langle \Lambda _{\widehat{f } } , \psi _{\varepsilon } \rangle \rightarrow \langle \Lambda _{\widehat{f } } , \psi \rangle . \) Since also \(\langle \delta _0 ^{(\alpha ) } , \psi _{\varepsilon } \rangle \rightarrow \langle \delta ^{(\alpha ) } _0 , \psi \rangle \) for all \(|\alpha | \le K \), the lemma follows. \(\square \)
The lemma immediately implies that if \(\psi \in \mathcal {S } ({\mathbb {R}}^n ) \), then \(\langle \widehat{f } * \chi _{\varepsilon } , \Sigma _h ' (\psi ) \rangle \rightarrow \langle \widehat{f } , \Sigma _h ' (\psi ) \rangle \), so \(\Sigma _h ( \widehat{f } * \chi _{\varepsilon } ) \rightarrow \Sigma _h (\widehat{f } ) \) in \(\mathcal {S }' ({\mathbb {R}}^n ) \) and even in \((C^{K , \lambda } ) ' \) with K and \(\lambda \) as above.
On the other hand, if we let \(f_{\varepsilon } \) be the inverse Fourier transform of \(\widehat{f } * \chi _{\varepsilon } \), then \(f_{\varepsilon } \in \mathcal {S } ({\mathbb {R}}^n ) \) since \(\widehat{f } * \chi _{\varepsilon } \) is, and \(\widehat{s_h [ f_{\varepsilon } ] } = \Sigma _h ( \widehat{f } * \chi _{\varepsilon } ) . \) We have that \(f_{\varepsilon } (x) = (2 \pi )^{n } f(x) \check{\chi } (\varepsilon x ) \), with \(\check{\chi } \) the inverse Fourier transform of \(\chi \), so \(\check{\chi } \in \mathcal {S }({\mathbb {R}}^n ) \) and \((2 \pi )^n \check{\chi } (0) = 1 . \) By hypotheses, \(f \in L^{\infty } _{- p } \) for some \(p < \kappa . \) If \(a > 0 \) such that \(p + a < \kappa \), then writing \(\widetilde{\chi } := (2 \pi )^n \check{\chi } \),
as \(\varepsilon \rightarrow 0 \), using for example the first order Taylor expansion for \(\widetilde{\chi } \) for \(|x | \le \varepsilon ^{- 1 / 2 } \) plus a trivial estimate for \(|x | > \varepsilon ^{-1 / 2 } . \) This certainly implies that \(s_h [ f_{\varepsilon } ] \rightarrow s_h [f ] \) in \(\mathcal {S }' ({\mathbb {R}}^n ) \), so we conclude that \(\widehat{s_h [f_{\varepsilon } ] } = \Sigma _h ( \widehat{f } * \chi _{\varepsilon } ) \rightarrow \widehat{s_h [f ] } \) and therefore \(\widehat{s_h [f ] } = \Sigma _h (\widehat{f } ) \) as distributions.
We finally show that \(\Sigma _h (\widehat{f } ) = \widehat{f } + F \), where F is the \(L^1 \)-function
We first check that F is well-defined and in \(L^1 \): first of all, each of the terms on the right hand side is in \(L^1 \), on account of the Fix–Strang conditions satisfied by for \(\widehat{L }_1 \) and the integrability of \((|\xi |^{\kappa } \wedge 1 ) \widehat{f }(\xi ) . \) Next, the function \((\xi , k ) \rightarrow \widehat{f }(\xi + 2 \pi h^{-1 } k ) (\widehat{L }_1 (h \xi ) - \delta _{0k } ) \) is absolutely integrable on \({\mathbb {R}}^n \times {\mathbb {Z}}^n \) with respect to the product of Lebesgue measure and the counting measure, since
Fubini’s theorem then implies that \(F(\xi ) \) is well-defined for almost all \(\xi \in {\mathbb {R}}^n \) and that \(F \in L^1 ({\mathbb {R}}^n ) . \) If \(\psi \in \mathcal {S } ({\mathbb {R}}^n ) \), then a double application of Fubini will show that
Since, by the Fix–Strang conditions (10), all derivatives of order \(\le \lceil \kappa \rceil - 1 \) of \(\Sigma _h (\psi ) - \psi \) in 0 are 0, the last integral is equal to \(\langle \widehat{f } , \Sigma _h ' (\psi ) - \psi \rangle = \langle \Sigma _h (\widehat{f } )- \widehat{f } , \psi \rangle \), and therefore \(\Sigma _h (\widehat{f } ) - \widehat{f } = F \), which finishes the proof of lemma 3.5.
Remark B.2
The lemma and its proof generalizes to f’s such that \(\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } \) is a finite Borel measure with respect to which the function \(|\xi |^{\kappa } \wedge 1 \) is integrable, provided that \(\kappa \notin {\mathbb {N}} \) (the reason being that we then no longer have (101)).
1.2 Appendix B.2: Proof of Lemma 5.4
It again suffices to consider the case of compactly supported \(\widehat{f } \)’s. We use the notations of the proof of lemma 3.5 above: in particular, let \(f_{\varepsilon } \) be the inverse Fourier transform of \(\widehat{f } * \chi _{\varepsilon } \) where \(\chi _{\varepsilon } = \varepsilon ^{-n } \chi ( \cdot / \varepsilon ) \) is an approximation of the identity. We have seen that \(\Sigma _h (\widehat{f }_{\varepsilon } ) \rightarrow \Sigma _h (\widehat{f } ) \) in \(\left( C^{K , \lambda } \right) ' \), where \(K = \lceil \kappa \rceil - 1 \) and \(\lambda = \kappa - K . \) Since \(e^{- h^{-2 } t G (h \cdot ) } \in C^{K , \lambda } \) this implies that
in \(\left( C^{K , \lambda } \right) ' \) and hence in \(\mathcal {S }' ({\mathbb {R}}^n ) . \)
On the other hand, we have seen in the proof of lemma 3.5 that \(f_{\varepsilon } \rightarrow f \) in \(L^{\infty } _{- p - a } \) if \(a > 0 . \) Hence by Lemma 5.1, if \(a < \kappa - p \) then \(u_h [f_{\varepsilon } ] \rightarrow u_h [f ] \) in \(L^{\infty } _{- p - a } \) and therefore as tempered distributions. This implies that
where we used Lemma 5.2. Hence \(\widehat{u_h [f ] } = e^{- h^{-2 } t G (h \cdot ) } \Sigma _h (\widehat{f } ) = e^{- h^{-2 } t G (h \cdot ) } \widehat{s_h [f ] } \) as tempered distributions, as claimed. We finally prove (51): if we let
then g is a \(C^{K , \lambda } \) - function of \(\xi \) and
Since \(g (\xi , t ; h ) \) vanishes to order \(|\xi |^{\kappa } \) in \(\xi = 0 \), by Proposition 5.3(ii), the representation \(\widehat{f } = \Lambda _{\widehat{f } }+ \sum _{|\alpha | \le \lceil \kappa \rceil - 1 } c_{\alpha } \delta ^{(\alpha ) } \) from the proof of lemma 3.5 shows that the distribution \(g (\cdot , t , h ) \widehat{f } \) can be identified with the locally integrable function \(\xi \rightarrow g(\xi , t , h ) \widehat{f }(\xi ) . \) \(\square \)
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Baxter, B., Brummelhuis, R. Convergence Estimates for Stationary Radial Basis Function Interpolation and for Semi-discrete Collocation-Schemes. J Fourier Anal Appl 28, 53 (2022). https://doi.org/10.1007/s00041-022-09945-3
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DOI: https://doi.org/10.1007/s00041-022-09945-3
Keywords
- Radial basis functions
- Stationary interpolation
- Semi-discrete collocation schemes (method of lines)
- Convergence estimates
- Approximate approximation
- Evolution equations
- Pseudo-differential operators
- Kolmogorov forward equations for Lévy processes