Abstract
Estimates of best approximations by exponential type analytic functions in Gaussian random variables with respect to the Malliavin derivative in the form of Bernstein–Jackson inequalities with exact constants are established. Formulas for constants are expressed through basic parameters of approximation spaces. The relationship between approximation Gaussian Hilbert spaces and classic Besov spaces are shown.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and Main Results
As is known (see [2, 7, 10, 26, 27]), the best approximations by differentiable functions in the classic analysis is based on a concept of the E-functional which characterizes the rapidity of approximations. This approach is constructive because it combines approximations with interpolation methods that provide explicit formulas for evaluating the approximations. In this area, many important inverse and direct theorems in the form of Bernstein–Jackson inequalities have been proven, in particular, in [3, 5, 11, 18, 19]. But approximation constants were not calculated that gives only asymptotic estimates of errors.
Our goal is to extend inverse and direct theorems in the form of Bernstein–Jackson inequalities on a more general case of best approximations by entire analytic in a Malliavin sense functions of random variables on Gaussian Hilbert spaces, and furthermore to calculate the explicit formulas for exact approximation constants in these inequalities.
The main results are presented in Sect. 3. Namely, in Theorem 2 it is established the bilateral version of Bernstein–Jackson inequalities
where the approximation Gaussian Hilbert space \(E_{\theta ,q}(\mathscr {E}^{p_0},L^{p_1})\) is represented as a fractional power of the real interpolation space \(K_{\theta ,q}\left( \mathscr {E}^{p_0},L^{p_1}\right) \) in the form
which is a generalization of the known classic isomorphism (see e.g. [2, Theorem 7.1.7]) on the case of Gaussian Hilbert spaces.
One of main results in Theorem 2 is also the explicit formula (12) for the best approximation constants \({C}_{\theta ,q}\) which for the case \(q=2\) receives the following simple form
The above-mentioned Bernstein–Jackson inequalities two-sided characterize the rapidity of approximations in the space \(L^{p_1}(\Omega ,\mathcal {F},{P})\) by the dense quasi-normed subspace \(\left( \mathscr {E}^{p_0},\vert \cdot \vert _{\mathscr {E}^p}\right) \) of convergent exponential types power series with respect to the Malliavin derivative \(\nabla \),
More specific, we consider Gaussian Hilbert spaces of random variables \(\phi _h\) defined on a complete probability space \((\Omega ,\mathcal {F},{P})\) such that \(\phi _h\sim \textrm{N}(0,\Vert h\Vert _H^2)\), where h belongs to a separable real Hilbert space H and \(\sigma \)-field \(\mathcal {F}\) is generated by a Gaussian field \(H\ni h\mapsto \phi _h\). This means that \(\phi _h\) is a family of Gaussian random variables with covariance structure \(\mathop {\textsf{E}}\phi _h\phi _g={\langle {h}\mid {g}\rangle }\), where \(\mathop {\textsf{E}}\phi _h\) is the expectation of \(\phi _h\) relative to \((\Omega ,\mathcal {F},{P})\) (see e.g. [15, Theorem 1.23]).
One of the main tools that is used to characterize on Gaussian Hilbert spaces of entire analytic functions is the notion of an exponential type, introduced in Sect. 2. This notion is a generalization of analytic vectors in the Nelson sense [21] for an abstract linear unbounded operator on the case of Malliavin’s derivative \(\nabla \).
Note that the case of non-stochastic entire analytic vectors in the Nelson sense were early analyzed in [8, 9].
The following properties of exponential-type random Gaussian variables are proved in Theorem 1. For this purpose, we introduce the quasi-normed space
of entire analytic functions in random variables of all exponential types \(\nu >0\), which is dense in \(L^p(\Omega ,\mathcal {F},{P})\) with \({p\in (0,\infty )}\) and the interpolation couple
which is compatible in the interpolation theory sense. Using these spaces, we define the best approximation E-functional to be
It is proved that each restriction \(\nabla \vert _{\mathscr {E}^{\nu ,p}}\) to the subspace \(\mathscr {E}^{\nu ,p}(\Omega ,\mathcal {F},{P})\) with a fixed exponential type \(\nu >0\) has the finite norm \(\le \nu \) and that \(\mathscr {E}^p(\Omega ,\mathcal {F},{P})\) and \(\mathscr {E}^{\nu ,p}(\Omega ,\mathcal {F},{P})\) are complete. The completeness is proved with the help of Bernstein compactness theorem for entire analytic functions of an exponential type [22, Theorem 3.3.6].
Notice additionally that in the considered case, the interpolation Gaussian Hilbert space \(K_{\theta ,q}\left( \mathscr {E}^{p_0},L^{p_1}\right) \) is determined through the quadratically modified (adapted to the case of Hilbert spaces) form of the K-functional, which was used, in particular, in [17].
Finally, the approach developed in this work naturally includes the case of functions with independent random variables, defined on infinite dimensional Banach spaces (see Example 1).
In Example 2 it is also shown that for the Gaussian space \(L^p\left( \mathbb {R}^d,\mathcal {F},\gamma _d\right) \) with \(p\in (1,\infty )\), endowed with the gaussian measure \(\gamma _d\) on the Borel \(\sigma \)-field \(\mathcal {F}=\mathcal {B}(\mathbb {R}^d)\), the previous approximation space has the form
where the space \(B_{p,\tau }^{s}(\mathbb {R}^d)\) with \(s=-1+1/\theta \) and \(\tau =q\theta \) exactly coincides with the classic Besov space (see e.g. [33, p. 197]). Above, the element \(f_g=g(\phi _{h_1},\ldots ,\phi _{h_d})\in \mathscr {E}^{p}\left( \mathbb {R}^d,\mathcal {F},\gamma _d\right) \) means the cylindrical random function determined by an entire analytic function \(g(z_1,\ldots ,z_d)\) on \(\mathbb {C}^d\) of an exponential type. In this case for \(q=2\) and \(\tau =2\theta \) the Bernstein–Jackson inequalities take the form
It is important to note that among other widely known universal approaches to approximating functions in Gaussian variables, the known Stein method [31, 32] and its subsequent modifications should be specially recorded. The following publications of recent years [12, 13, 23, 24, 30] are devoted to the development of these studies using the Malliavin calculus.
2 Exponential Type with Respect to the Malliavin Derivative
Let a real separable Hilbert space H with scalar product and norm, denoted by \(\langle \cdot \mid \cdot \rangle \) and \(\Vert \cdot \Vert _H=\langle \cdot \mid \cdot \rangle ^{1/2}\), has an orthonormal basis \(\left\{ \mathfrak {e}_i:i\in \mathbb {N}\right\} \).
There exists a linear isometry \(H\ni h\rightarrow \phi _h\) into a Gaussian Hilbert space of real-valued functions defined on a complete probability space \((\Omega ,\mathcal {F},{P})\), where the \(\sigma \)-field \(\mathcal {F}\) is generated by H (see e.g. [15, Theorem 1.23]). We suppose that \(\phi _h\) is centered and has the covariance \(\mathop {\textsf{E}}\phi _h\phi _g={\langle {h}\mid {g}\rangle _H}\) (see [25, no 1.1]).
The \(L^p\)-norm of real-valued functions f on \((\Omega ,\mathcal {F},{P})\) is defined by
All \(L^p\)-norms with \(p\in (0,\infty )\) are proportional [15, Theorem 1.4], since \(\Vert f\Vert _p=\kappa (p)\Vert f\Vert _2\), where \({\kappa (p)=\sqrt{2}(\varGamma ((p+1)/2)/\sqrt{\pi })^{1/p}}\). By definition, the space \(L^p=L^p(\Omega ,\mathcal {F},{P})\) is endowed with the \(L^p\)-norm. The space of all measurable functions \(L^0=L^0(\Omega ,\mathcal {F},{P})\) is equipped with the topology of convergence in probability, metrizable by \(\Vert f\Vert _0=\mathop {\textsf{E}}\min (\vert f\vert ,1)\).
The \(L^p\)-norm of Y-valued functions \(\xi =f\otimes y\) in \((\Omega ,\mathcal {F},{P})\) is defined to be \(\Vert \xi \Vert _p=\left\{ \begin{array}{ll} (\mathop {\textsf{E}}\Vert \xi \Vert _Y^p)^{1/p}&{}\hbox {if }p\in (0,\infty )\\ \mathop {\mathrm {ess\,sup}}\Vert \xi \Vert _Y&{}\hbox {if }p=\infty \end{array}\right. \), where y belongs to a Banach space \((Y,\Vert \cdot \Vert _Y)\). Let \(L^p(Y)\) be the completion of linear span of \(\xi =\phi \otimes y\) with respect to this \(L^p\)-norm.
Consider the class of smooth functions of cylindrical forms
where \(\phi _{h_1},\ldots ,\phi _{h_n}\in L^0\) with \(h_i\in H\) and \(F\in C^\infty _b(\mathbb {R}^n)\) is a smooth function with bounded partial derivatives \(\partial _i\). By definition the Malliavin derivative \(\nabla \) of f is the H-valued random variable
(see e.g. [25, no 1.2.1]). In particular, \(\nabla \phi _h=h\) for every \(h\in H\).
For \(p\in [1,\infty )\) the domain \(W^{1,p}\) of \(\nabla \) is the closure in \(L^p\) of all functions with respect to the graph-norm
The completion of linear spans of tensor products \(\psi _n={h_1\otimes \ldots \otimes h_n}\), \({(h_\imath \in H)}\) endowed with \(\Vert \psi _n\Vert _{{H}^{\otimes n}}={\langle \psi _n\mid \psi _n\rangle ^{1/2}}\) is denoted by \({H}^{\otimes n}\), where \({\langle \psi _n\mid \psi _n'\rangle }= {\langle h_1\mid h'_1\rangle \ldots \langle h_n\mid h'_n\rangle }\). Let \(h^{\otimes n}:={h\otimes \ldots \otimes h}\). The symmetric tensor power \(H^{\odot n}\subset H^{\otimes n}\) is defined to be a range of the orthogonal projector
where \(S_n\) means n-elements permutations. The corresponding symmetric Fock space \(\varGamma (H)=\bigoplus _0^\infty {H}^{\odot n}\) of elements \(\psi =\bigoplus \psi _n\) with \(\psi _n\in H^{\odot n}\) and \(H^{\odot 0}=\mathbb {R}\) is endowed with the norm
The iterated derivative \(\nabla ^kf\) with \(k>1\) is a random variable with values in \(H^{\odot k}\). Its domain \(W^{k,p}\) coincides with the closure of Malliavin-smooth random variables with respect to the graph-norm
The operators \(\nabla ^k:W^{k,p}\rightarrow L^p(H^{\odot k})\) are closed and \(\bigcap _{k=0}^{\infty }W^{k,p}\) is dense in \(L^p(\Omega ,\mathcal {F},P)\) because one contains all Hermite polynomials (see e.g. [25, 1.5]).
Definition 1
A function \(f\in \bigcap _{k=0}^{\infty }W^{k,p}\) with \({p\in (1,\infty )}\) of Gaussian random variables on \((\Omega ,\mathcal {F},{P})\) we call the exponential type \(\nu >0\) with respect to the Malliavin derivative \(\nabla \) if the power series
is an entire analytic function in the complex variable \(z\in \mathbb {C}\) of the exponential type \(\nu \), that is, for which the following condition is satisfied (see, e.g. [4, Theorem 1.1.1]),
Definition 1 can be considered as a generalization of analytic vectors for a linear unbounded operator in the sense of Nelson (see [21]) on the case of derivative \(\nabla \). As we will see below, the series
is pointwise absolutely convergent on a non-trivial dense subspace of \(L^p\).
Definition 2
-
(i)
Let \(\mathscr {E}^{\nu ,p}\) with \({p\in (1,\infty )}\) be the subspace in \(L^p\) of functions f in random variables with the finite Hilbertian norm
$$\begin{aligned} \Vert f\Vert _{\mathscr {E}^{\nu ,p}}=\bigg (\sum _{k=0}^\infty \frac{1}{\nu ^{2k}} \left( \mathop {\textsf{E}}\Vert \nabla ^kf\Vert ^p_{H^{\odot k}}\right) ^{2/p}\bigg )^{1/2},\quad {\nu >0}. \end{aligned}$$(3) -
(ii)
Let the subspace of functions f in \(L^p\),
$$\begin{aligned} \mathscr {E}^p=\bigcup _{\nu >0}\mathscr {E}^{\nu ,p}, \end{aligned}$$is endowed with the quasi-norm
$$\begin{aligned} \vert f\vert _{\mathscr {E}^p}=\Vert f\Vert _p+\inf \big \{\nu >0:f\in \mathscr {E}^{\nu ,p}\big \}. \end{aligned}$$(4)
Theorem 1
-
(a)
The spaces \((\mathscr {E}^{\nu ,p},\Vert \cdot \Vert _{\mathscr {E}^{\nu ,p}})\) and \((\mathscr {E}^p,\vert \cdot \vert _{\mathscr {E}^p})\) are complete.
-
(b)
Each restriction \(\nabla \vert _{\mathscr {E}^{\nu ,p}}\) is a linear operator with a finite norm \(\le \nu \) on the space \({\left( \mathscr {E}^{\nu ,p},\Vert \cdot \Vert _{\mathscr {E}^{\nu ,p}}\right) }\). The following contractive inclusions hold,
$$\begin{aligned} \mathscr {E}^{\nu ,p}\looparrowright \mathscr {E}^{\mu ,p}\looparrowright L^p,\quad {\mu>\nu >1}. \end{aligned}$$(5) -
(c)
The space \(\mathscr {E}^{p}\) with \(p\in (1,\infty )\) is dense in \(L^{q}\) for any \({q\in (0,\infty )}\).
-
(d)
The interpolation couple \((\mathscr {E}^{p_0},L^{p_1})\) for any \(p_0\in (1,\infty )\) and \(p_1\in (0,\infty )\) is compatible.
Proof
(a) Check that \(\vert \cdot \vert _{\mathscr {E}^p}\) is a quasi-norm. For any \({f\in \mathscr {E}^{t,p}}\) and \({g\in \mathscr {E}^{s,p}}\),
This in particular ensures that \(\mathscr {E}^p\) is a quasi-normed linear subspace.
Prove the completeness of the space \(\mathscr {E}^{\nu ,p}\). Let \((f_n)\) be a fundamental sequence in \(\mathscr {E}^{\nu ,p}\), i.e.,
From the representation (3) for \(\Vert \cdot \Vert _{\mathscr {E}^{\nu ,p}}^2\) as a sum of positive addends, it follows that the sequences \((f_n)\) and \(\big (\nabla ^kf_n/\nu ^k\big )\) with \({k\ge 1}\) are fundamental in \(L^p\) and \(L^p(H^{\odot k})\), respectively.
Hence, there are elements \({f\in L^p}\) and \({g_k\in L^p(H^{\odot k})}\) such that \({f_n\rightarrow f}\) in \(L^p\) and \({\nabla ^k f_n/\nu ^k\rightarrow g_k}\) in \(L^p(H^{\odot k})\) for any \({k\ge 1}\). By closeness of \(\nabla ^k\), the equality \({g_k=\nabla ^kf/\nu ^k}\) holds, i.e.,
Since \(\Vert f_n\Vert _{\mathscr {E}^{\nu ,p}}\le \Vert f_n-f_{n_\varepsilon }\Vert _{\mathscr {E}^{\nu ,p}}+ \Vert f_{n_\varepsilon }\Vert _{\mathscr {E}^{\nu ,p}}\le \varepsilon +\Vert f_{n_\varepsilon }\Vert _{\mathscr {E}^{\nu ,p}}\) for all \(n\ge n_{\varepsilon }\), we find \(\Vert f\Vert _{\mathscr {E}^{\nu ,p}}\le \varepsilon +\Vert f_{n_\varepsilon }\Vert _{\mathscr {E}^{\nu ,p}}\) by taking the limit in \(L^p\) as \(n\rightarrow \infty \). As a result, \(f\in \mathscr {E}^{\nu ,p}\), since \(f_{n_\varepsilon }\in \mathscr {E}^{\nu ,p}\). Thus, \(\mathscr {E}^{\nu ,p}\) is complete.
Further, we note that the Laplace transform of a function (2) has the form
Hence, for the norm in \(\mathscr {E}^{\nu ,p}\), we get the integral representation
Let now \((f_n)\) be a fundamental sequence in the quasi-normed space \(\mathscr {E}^p\). Hence, there exists \({\nu >0}\) such that \({\vert f_n\vert _{\mathscr {E}^p}<\nu }\) for all \({n\in \mathbb {N}}\) thus
It means that \({(f_n)\subset \mathscr {E}^{\nu ,p}}\). Consider the restriction to \(\mathbb {R}\) of the correspondent sequence of complex entire functions \((\hat{f}_n)\) of an exponential type \(\nu \), defined by (2). By (8), the following sequence is bounded by a constant \({K_\nu >0}\),
Hence, in accordance with Bernstein’s compactness theorem [22, Theorem 3.3.6] there exists a convergent subsequence \(\left\{ (\hat{f}_{n_i}-\hat{f}_{m_i})(t)\exp (-t\nu ^2) :i\in \mathbb {N}\right\} \) with respect to the uniform convergence in the variable \(t\in [0,r]\) for any \({r>0}\).
Thus, \({\forall \varepsilon >0}\), \(\exists {n_\varepsilon \in \mathbb {N}}:\)
where \(r=r_\varepsilon \) is chosen large enough that \(K_\nu \exp {(-r_\varepsilon \nu ^2)}<\varepsilon \). Using the integral representation (7), we obtain
for all \(n_i,m_i\ge n_\varepsilon \). As a result, \((f_{n_i})\) is fundamental in \(\mathscr {E}^{2\nu ,p}\). According to the completeness of \(\mathscr {E}^{2\nu ,p}\), there exists an element \({f\in \mathscr {E}^{2\nu ,p}}\) such that \({f_{n_i}\rightarrow f}\) as \(i\rightarrow \infty \). Thus \(\mathscr {E}^p\) is complete.
(b) First note that according to the known classical formula (see e.g. [4, Theorem 1.1.1]), the function (2) has the exponential type \(\nu ^2\) if and only if its Laplace transform (6) satisfies the following condition
It follows, in particular, that the formula (3) defines the norm on the space \(\mathscr {E}^{\nu ,p}\) correctly. Moreover, using that for every \(f\in \mathscr {E}^{\nu ,p}\) the inequality
holds, the restriction \(\nabla \vert _{\mathscr {E}^{\nu ,p}}\) is a bounded operator with a norm \(\le \nu \). The recursive reasoning gives \(\Vert \nabla ^kf\Vert _{\mathscr {E}^{\nu ,p}}\le \nu ^k\Vert f\Vert _{\mathscr {E}^{\nu ,p}}\) for all \({k\ge 0}\). It follows that
Thus, for \(\mu >\nu \) the following convergent series satisfies the inequality
that give the inclusions (5) for \(\mu >\nu \).
(c) Consider the Gaussian exponential defined for random variables \(\phi _h\),
As is known (see [15, Theorem 3.33]), the corresponding exponential series is convergent in \(L^2\) thus in \(L^p\) for \(p\in (0,\infty )\). The equality
follows from the property
since the expression \(\exp (-\mathop {\textsf{E}}\phi ^2_h/2)\) does not depend on all \(g\in H\). Hence,
for any \(t\in \mathbb {R}\), where the tensor exponential series \(\exp (h)\) is convergent in the symmetric Fock space \(\varGamma (H)\). Moreover, from the formula (1) for norm in \(\varGamma (H)\) it follows \(\Vert \exp (h)\Vert _\varGamma =\exp \Vert h\Vert \). So, for \(f=\mathscr {G}_h\), we have the representation
Applying the formula (9) to this power series, we get that \(f=\mathscr {G}_h\) has the following exponential type
Hence, \(\mathscr {G}_h\in \mathscr {E}^{\Vert h\Vert ,p}\) for any \(h\in H\).
On the other side, it is known (see [15, Theorem 2.12 & Corollary 3.40]) that
is total in \(L^q\) for any \(q\in (0,\infty )\), where \(\{\mathfrak {e}_i\}\) is an orthogonal basis in H. More specific, it follows from the fact that the family of all Hermite polynomials \(\mathfrak {h}_n\) in random variables \(\left\{ \mathfrak {h}_n(\phi _h):{h\in H}, n\in \mathbb {N}\cup \{0\}\right\} \) is total \(L^q\) for any \(q\in (0,\infty )\), since \(L^q\)-norms are proportional to the \(L^2\)-norm (see [15, Theorem 1.4]). As a result, the subspace
with \(p\in (1,\infty )\) is dense in \(L^q\) for any \(q\in (0,\infty )\).
(d) This statement is a direct conclusion of (c). In fact, the couple quasi-normed spaces \((\mathscr {E}^{p_0},L^{p_1})\) on the same \((\Omega ,\mathcal {F},{P})\) can be consider as a dense subspace in the algebraic sum of spaces \(L^{p_0}+L^{p_1}\) endowed with the quasi-norm
which guarantees the compatibility (see e.g. [2, Lemma 3.10.3] or [16, no 1]). \(\square \)
3 Exact Estimates of Best Approximations on Gaussian Hilbert Spaces
In what follows, our goal is to prove the inverse and direct approximation theorems on Gaussian Hilbert spaces by Malliavin-entire functions of random variables in the form of Bernstein–Jackson inequalities with exact constants.
Given the compatible interpolation couple of quasi-normed Gaussian spaces
we define the best approximation E-functional
where \(f=f_0+f_1\) belongs to the algebraic sum \({\mathscr {E}^{p_0}+L^{p_1}}\) such that \({f_0\in \mathscr {E}^{p_0}}\) and \({f_1\in L^{p_1}}\). For any pairs indexes
the corresponding best approximation scale is defined to be the following scale of quasi-normed Gaussian spaces
It is natural to call the space \(E_{\theta ,q}(\mathscr {E}^{p_0},L^{p_1})\) approximation for the compatible interpolation couple \((\mathscr {E}^{p_0},L^{p_1})\) on the same probability space \((\Omega ,\mathcal {F},{P})\).
In what follows, we will prove that the approximation constants
determined by the normalization factor of Lions–Peetre’s interpolation method,
are exact for both Bernstein–Jackson inequalities. Note that \(N_{\theta ,q}=N_{1-\theta ,q}\) (see e.g. [17, p. 99]). The approximation constant for \(q=2\) receives the form
In fact, by integrating the above functions (see e.g. [20, Example B.5, Theorem B.7] or [17, p. 99]) it follows that for \(q=2\) the normalization factor (14) in the interpolation K-method employed here is equal to \(N_{\theta ,2}=\left( {2\sin \pi \theta }/{\pi }\right) ^{1/2}.\)
The following approximation theorem is based on analytical properties of an exponential type of Gaussian random variables with respect to the Malliavin derivative which were established in Theorem 1.
Theorem 2
-
(a)
The Bernstein–Jackson bilateral inequalities
$$\begin{aligned} t^{-1+1/\theta }E(t,f)&\le {C}_{\theta ,q}\Vert f\Vert _{E_{\theta ,q}}\le 2^{1/2\theta }\vert f\vert ^{-1+1/\theta }_{\mathscr {E}^{p_0}} \Vert f\Vert _{p_1} \end{aligned}$$(15)with the approximation constant (12) for all \(f\in \mathscr {E}^{p_0}\cap L^{p_1}\) hold.
-
(b)
The following isomorphism is valid up to norm equivalence,
$$\begin{aligned} E_{\theta ,q}(\mathscr {E}^{p_0},L^{p_1})\simeq K_{\theta ,q}\left( \mathscr {E}^{p_0},L^{p_1}\right) ^{1/\theta }. \end{aligned}$$(16) -
(c)
There is a unique extension of the left inequality in (15) to the following Jackson-type inequality on the whole Gaussian approximative space
$$\begin{aligned} E(t,f)&\le t^{1-1/\theta }C_{\theta ,q}\Vert f\Vert _{E_{\theta ,q}}\quad \text {for all}\quad {f\in E_{\theta ,q}(\mathscr {E}^{p_0},L^{p_1})}. \end{aligned}$$(17)
Proof
(a) We will use the classical integral of the Lions–Peetre real interpolation method
Consider the quadratic K-functional (see e.g. [17] or [20, App. B]) for the interpolation couple of quasi-normed spaces \((\mathscr {E}^{p_0},L^{p_1})\) with \({p_0\in [1,\infty )}\) and \({p_1\in (0,\infty )}\),
determining the real interpolation space (both alternative notations are used),
which is endowed with the norm
First, let \(0<q<\infty \). By integration both sides of the following inequality
we successively find
After summing up, it follows the inequality
Let \(K_\infty (t,f):=\inf _{f=f_0+f_1}\max \big \{\vert f_0\vert _{\mathscr {E}^{p_0}},t\Vert f_1\Vert _{p_1}\big \}\). It is easy to see that
By [2, Lemma 7.1.2] for every \(t>0\) there exists \(v>0\) such that
It follows that
Integrating by parts with the change of variables \(v=t/E(t,f)\) and using the known properties of functionals that \(v^{-\theta }K_\infty (v,f)\rightarrow 0\) as \({v\rightarrow 0}\) or \({v\rightarrow \infty }\) and \({t^{-1+1/\theta }E(t,f)\rightarrow 0}\) as \({t\rightarrow 0}\) or \({t\rightarrow \infty }\) (see [2, Theorem 7.1.7]), we get
Therefore, according to the first inequality (20) and the notation (11),
On the other hand, from the second inequality (20) it follows
Taking the root and combining the previous inequalities, we get
As a result, we obtain the isomorphism (16), which proves the claim (b), i.e., that
Let \(\alpha =\vert f\vert _{\mathscr {E}^{p_0}}/\Vert f\Vert _{p_1}\). Since \(K(t,f)\le \min \big \{\vert f\vert _{\mathscr {E}^{p_0}},t\Vert f\Vert _{p_1}\big \}\), we find
Taking the root above, this can be rewritten as
where \([q\theta (1-\theta )]^{-1/q}=\Vert \min \{1,\cdot \}\Vert _{\theta ,q}\). Applying the integral (18) to the inequality \(2^{-1/2}\min \{1,\cdot \}\le g(\cdot )\), we find \(N_{\theta ,q}\le 2^{1/2}\Vert \min \{1,\cdot \}\Vert _{\theta ,q}^{-1}\). Hence,
Now, by the second inequality in (23) and (12),(24), we find that
On the other hand, by (22), we have
Combining the last inequalities, we get the desired inequality (15).
Let us consider the case \(q =\infty \). Denote \(\alpha =\Vert f\Vert _{p_1}/\vert f\vert _{\mathscr {E}^{p_0}}\) with a nonzero element \(f\in \mathscr {E}^{p_0}\cap L^{p_1}\). Since
or otherwise \(K(t,f)\le \vert f\vert _{\mathscr {E}^{p_0}}\min (1,\alpha t) ={\min \left( \vert f\vert _{\mathscr {E}^{p_0}},t\Vert f\Vert _{p_1}\right) },\) we get
Taking \(t=\vert f\vert _{\mathscr {E}^{p_0}}/\Vert f\Vert _{p_1}\), we obtain
So, the right side inequality in (15) holds. On the other hand,
Thus, the inequality (17) also is valid for this case.
(c) By Theorem 1, \(\mathscr {E}^{p_0}\) is complete, so \(K_{\theta ,q}\left( \mathscr {E}^{p_0},L^{p_1}\right) \) is complete, as interpolation of complete spaces. \(\square \)
Remark 1
The relationship between the weight function \(g(t)=t^2/(1+t^2)\) and the square K-functional, and therefore also the E-functional, is explained by the formula
[20, Example B.4]. It follows from \(\min _{z=z_0+z_1}\left( \alpha _0\vert z_0\vert ^2+\alpha _1\vert z_1\vert ^2\right) ={\alpha _0\alpha _1\vert z\vert ^2}{\alpha _0+\alpha _1}\) for a fixed \(\alpha _0,\alpha _1>0\) and a complex z. This minimum is achieved when \(\alpha _0z_0=\alpha _1z_1={\alpha _0\alpha _1z}/{(\alpha _0+\alpha _1)}.\) Thus, K(t, 1) is minimized when \(f_0\), \(f_1\) are such that
For the space \(L^2(\Omega ,\mathcal {F},{P})\) previous results can be made more specific.
Corollary 3
On the space \(E_{\theta ,2}(\mathscr {E}^2,L^2)\) endowed with the quasi-norm
defined by the best approximation E-functional
the following Bernstein–Jackson type inequalities are satisfied,
Proof
The inequalities (25) - (26) directly follow from Theorem 2 (a,c). \(\square \)
Corollary 4
The norm on the Hilbert space \(\mathscr {E}^{\nu ,2}\) satisfies the equality
where \(\Vert \cdot \Vert _{\mathcal {H}^2(D_\nu )}\) in (27) is the Hilbertian norm for analytic functions
belonging to the Hardy space \(\mathcal {H}^2(D_\nu )\). Herewith, the isometric isomorphism
determined by the linear mapping \(f\longmapsto \hat{F}\), holds.
Proof
The isometry (27) follows from the properties (2) and (7) of the Laplace transform \(\textsf{L}\) for entire analytic functions, as well as, from the elementary fact that for every power series \(\hat{F}(z)=\sum {c}_kz^k\) from \(\mathcal {H}^2(D_\nu )\) its norm satisfies the equality
The isometric equation (28) is a consequence of the equality (7) for the norm \({\Vert \cdot \Vert _{\mathscr {E}^{\nu ,2}}}\). \(\square \)
Corollary 5
The quasi-norm on \({(\mathscr {E}^2,\vert \cdot \vert _{\mathscr {E}^2})}\) admits the representation
Moreover, \(\mathscr {E}^2\) has also a stronger nuclear topology of the inductive limits
with compact inclusions.
Proof
The proof of (29) follows from the known formula (9) [4, Theorem 1.1.1] for Taylor coefficients of complex entire analytic functions of an exponential type, expressed through its Laplace-image (see, the proof of Theorem 1(b)).
The compactness of inclusions (5) is proved Theorem 1(c) based on Bernstein’s compactness theorem [22, Theorem 3.3.6]. Nuclearity of inductive limits with compact inclusions are a well-known fact (see e.g. [29, no 7.4]). \(\square \)
Corollary 6
The 1-parameter family of linear operators \(T_s:L^2\rightarrow L^2(\varGamma (H))\), uniquely defined by the mapping
satisfies the following invariant property
where the number operator \(\mathcal {N}\) is determined on the Fock space \(\varGamma (H)\). Moreover, the derivative \(\nabla :W^{1,2}\rightarrow L^2(H)\) coincides with a universal annihilator of \(T_s\), i.e.,
Proof
From the proof of Theorem 1(c), we directly get
and, as a consequence, (31). On the other hand, \(\mathcal {N}\) is the infinitesimal generator of the 1-parameter second-quantization semigroup \(\varGamma (e^{-s I_H})=e^{-s\nabla }\) with the identical operator \(I_H\) on H (see [25, no 1.4]), thus
It follows the equality (30). The uniqueness of extension \(T_s\) onto \(L^2\) is due to totality \(\mathscr {G}_h\) in \(L^2\) and \(\mathscr {G}_h\otimes \exp (h)\) in \(L^2(\varGamma (H))\).
Moreover, according to [1, no 3], the derivative \(\nabla \) is a universal annihilator and the equality (32) is valid. \(\square \)
4 Application Examples
Example 1
Let us consider the space \(L^p(X,\mathcal {F},\gamma )\) with \(1\le p\le \infty \) of functions f in Gaussian random variables \(X\ni x\mapsto \phi _h(x)\) for all \(h\in {H}\), defined on the probability space \((X,\mathcal {F},\gamma )\) over an abstract Wiener space (X, H) in the sense of Gross’s theory [14]. Here, let X be a separable real Banach space, \(H\subset X\) is a Cameron-Martin type reproducing kernel subspace, \(\mathcal {F}=\mathcal {B}(X)\) is the Borel \(\sigma \)-field on X and, in addition, the probability measure \(\gamma \) on \(\mathcal {F}\) is characterized by the property
The measure \(\gamma \) is Gaussian in the sense that each continuous linear functional \(x^*\in X^*\), regarded as a random variable \(x\mapsto x^*(x)\) on \((X,\mathcal {F},\gamma )\), is Gaussian. The expectation for this case is defined to be
for all \(p\ge 1\), where \(L^p\subset L^1\) because \(\gamma (X)=1\) (see e.g. [34, Theorem 2]).
In this case, the interpolation structure of the approximative Gaussian space is described by the isomorphism
where \(\{{0<\theta<1}, \ {0< q<\infty }\}\) or \(\{{0<\theta \le 1}, \ {q=\infty }\}\). According to Theorem 2 and Corrolary 3 the Bernstein–Jackson inequalities take the form
with the exact approximation constant \({C}_{\theta ,q}\) of the form (12), or (14) for the case \(p_0=p_1=2\).
Example 2
A special case of Example 1 is obtained for \(X=(\mathbb {R}^d,\vert \cdot \vert )\). Consider the Banach space \(L^p=L^p\left( \mathbb {R}^d,\mathcal {F},\gamma _d\right) \) with \(p\in (1,\infty )\) and \(\mathcal {F}=\mathcal {B}(\mathbb {R}^d)\), which is just the space of measurable functions in random variables relative to the gaussian measure
Each function \(G\in L^p\) can be approximated by entire analytic functions g of an exponential type \({t>0}\) with restrictions to \(\mathbb {R}^d\) belonging to \(L^p\) (see e.g. [22]). The best approximations can be characterized by the functional
where the subspace \(\mathscr {E}^p={\bigcup }_{t>0}\mathscr {E}^{p,t}\) of \(L^p\) is endowed with the quasi-norm
defined using the support of the Fourier-image \(\hat{g}\) (see [2, no 7.2]). By Paley–Wiener theorem for entire analytic functions of an exponential type, this quasi-norm can be rewritten in the form (4) (see e.g. [9]).
Now, taking any cylindrical random function \(f_g=g(\phi _{h_1},\ldots ,\phi _{h_d})\) determined by functions g of an exponential type t and applying the formula (9), we get
i.e., \(f_g\in \mathscr {E}^{p,t}\). The subspace of all functions \(f_g\) with such g and any t is dense in \(\mathscr {E}^p\), since it contains all polynomials of the random variables \(\phi _{h_1},\ldots ,\phi _{h_d}\).
On the other hand, it is known that if the space \(\mathscr {E}^p\), consisting of all entire analytic functions g of an exponential type on \(\mathbb {C}^d\), is endowed with the quasi-norm (33) then the suitable approximation space \(E_{\theta ,q}(\mathscr {E}^{p},L^{p})\) exactly coincides with the classic Besov space denoted by
(see [33, p. 197]). Hence, the equality (16) from Theorem 2 may be rewritten in the form
Then the corresponding Bernstein–Jackson inequalities take the form
where the constant \({C}_{\theta ,q}\) has the form (12), or (14) for the case \(p=2\).
Remark 2
The last example also shows that the Gaussian space
with \(\{{0<\theta<1}, \ {0< q<\infty }\}\) or \(\{{0<\theta \le 1}, \ {q=\infty }\}\), which characterizes the best approximations in \(L^{p_1}=L^{p_1}(\Omega ,\mathcal {F},{P})\) with two-sided precision by entire analytic functions relative to the Malliavin derivative, are the closest generalization of Besov spaces on the case of functions in Gaussian random variables.
Significant new generalizations and connections between the approximation and Besov-type spaces in a wider context are presented in [13] (see also references therein).
Data Availability
This manuscript has no associated data.
References
Applebaum, D.: Universal Malliavin calculus in Fock and Lévy-Itô spaces. Commun. Stoch. Anal. 3(1), 119–141 (2009)
Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)
Bernd, C.: Inequalities of Bernstein–Jackson-type and the degree of compactness of operators in Banach spaces. Ann. Inst. Fourier (Grenoble) 35(3), 79–118 (1985)
Bieberbach, L.: Analytische Fortsetzung. Springer, Berlin (1955)
Butzer, P.L., Scherer, K.: Jackson and Bernstain-type inequalities for families of commutative operators in Banach spaces. J. Approx. Theory. 5, 308–342 (1972)
Cobos, F., Domínguez, O.: Approximation spaces, limiting interpolation and Besov spaces. J. Approx. Theory 189, 43–66 (2015)
Cwikel, M., Peetre, J., Sagher Y., Wallin H.: Function Spaces and Applications. Proceedings of the US-Swedish Seminar, Springer (1986)
Dmytryshyn, M., Lopushansky, O.: Bernstein–Jackson-type inequalities and Besov spaces associated with unbounded operators. J. Inequal. Appl. 2014, 105 (2014)
Dmytryshyn, M., Lopushansky, O.: On spectral approximations of unbounded operators. Complex Anal. Oper. Theory 13(8), 3659–3673 (2019)
Feichtinger, H.G., Fuhr, H., Pesenson, I.Z.: Geometric space-frequency analysis on manifolds. J. Fourier Anal. Appl. 22, 1294–1355 (2016)
Garrigós, G., Hernández, E.: Sharp Jackson and Bernstein inequalities for N-term approximation in sequence spaces with applications. Indiana Univ. Math. J. 53(6), 1739–1762 (2004)
Geiss, C., Geiss, S., Laukkarinen, E.: A note on Malliavin fractional smoothness for Lévy processes and approximation. Potential Anal. 39, 203–230 (2013)
Geiss, S., Ylinen, J.: Decoupling on the Wiener Space, related Besov Spaces, and applications to BSDEs. Mem. Amer. Math. Soc. 1335 (2021)
Gross, L.: Abstract Wiener spaces. In H.D. Doebner, ed. Proc. 5th Berkeley Symp. Math. Stat. and Probab. Part 1, vol. 2. Berkeley: Univ. California Press. 31–42 (1965)
Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)
Komatsu, N.: A general interpolation theorem of Marcinkiewicz type. Tôhoku Math. J. 33, 383–393 (1981)
Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, Cham (1972)
LinSen, X., JiaCheng, L., SenHua, L., DunYan, Y.: Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces. Sci. China Math. 52(3), 481–492 (2009)
Malyarenko, A.A.: Local properties of gaussian random fields on compact symmetric spaces and theorems of the Jackson–Bernstein type. Ukr. Math. J. 51, 66–75 (1999)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Nelson, E.: Analytical vectors. Ann. Math. 70(3), 572–615 (1959)
Nikolskii, S.: Approximation of functions of several variables and imbedding theorems. Springer, Berlin (1975)
Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Related Fields 145(1–2), 75–118 (2009)
Nourdin, I., Peccati, G., Réveillac, A.: Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 45–58 (2010)
Nualart, D.: The Malliavin calculus and related topics, II Springer, Berlin (2006)
Peetre, J., Sparr, G.: Interpolation of normed Abelian groups. Ann. Mat. Pura Appl. 92(1), 217–262 (1972)
Pesenson, I.Z.: Jackson-type inequality in Hilbert spaces and on homogeneous manifolds. Anal. Math. 48(4), 1153–1168 (2022)
Prestin, J., Savchuk, V.V., Shidlich, A.L.: Direct and inverse theorems on the approximation of 2- periodic functions by Taylor Abel Poisson operators. Ukr. Math. J. 69(5), 766–781 (2017)
Schaefer, H.H., Wolff, M.P.: Topological Vector Spaces. Springer, Berlin (1999)
Shih, H.H.: On Stein’s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces. J. Funct. Anal. 261, 1236–1283 (2011)
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. University California Press, California, pp. 583–602 (1972)
Stein, C.: Approximation Computation of Expectations. IMS Lecture Notes Monogr. Ser. Institute of Mathematical Statistics, Hayward (1986)
Triebel, H.: Interpolation Theory. Function Spaces. Differential Operators. North-Holland Publications, Amsterdam (1978)
Villani, A.: Another note on the inclusion \(L^p(\mu )\subset L^q(\mu )\). Amer. Math. Monthly 92(7), 485–487 (1985)
Acknowledgements
The author would like to thank two anonymous referees for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
There are no conflicts and potential competing of interest to disclose.
Additional information
Communicated by Hans G. Feichtinger.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lopushansky, O. Bernstein–Jackson Inequalities on Gaussian Hilbert Spaces. J Fourier Anal Appl 29, 58 (2023). https://doi.org/10.1007/s00041-023-10035-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-023-10035-1
Keywords
- Bernstein–Jackson inequalities
- Approximation by Gaussian random variables
- Best approximation constants