Abstract
We construct trigonometric polynomials that fast decrease towards \(\pm \pi \). We apply them to construct a trigonometric polynomial the derivative of which interpolates the derivative of a given \(2\pi \)-periodic function, at some prescribed distinct points in \([-\pi ,\pi )\), and vanishes at some other prescribed points in that interval. The construction requires that the function possesses derivatives where the interpolation is supposed to take place. Still, we are able to apply the result to trigonometric approximation of a \(2\pi \)-periodic piecewise algebraic polynomial which is merely continuous, while interpolating its derivative at some points (that, obviously, are not knots).
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1 Introduction and the main results
Let \(I:=[-\pi ,\pi ]\) and let C[a, b] and \(C^l[a,b]\) denote, respectively, the space of continuous functions and of l times continuously differentiable functions, and denote by \(\tilde{C}\) the space of \(2\pi \)-periodic continuous functions. As usual all spaces are equipped with the sup-norm, i.e., \(\Vert f\Vert _{[a,b]}:=\max _{x\in [a,b]}|f(x)|\) and \(\Vert f\Vert :=\max _{x\in \mathbb R}|f(x)|\), respectively.
For \(\beta \in \mathbb N\) and \(n\in \mathbb N\), let
be a Jackson-type kernel, where \(C_*(\beta )\le \gamma _{n,\beta }\le C^*(\beta )\) is a normalizing factor, so that J is a trigonometric polynomial of degree \(\beta (n-1)\), and
(see, e.g., [1, p. 204]). For the asymptotic behavior of \(\gamma _{n,\beta }\), see [3, Theorem 1].
Put \(h:=\pi /n\). Clearly,
and Bernstein’s inequality implies, for all \(\nu \in \mathbb N_0\),
In addition, we will show that (see the end of Sect. 2) for \(0\le \nu \le 2\beta \),
where \(C_1(\beta )\) and \(C_3(\beta )\) depend only on \(\beta \), and \(C_2(\beta ,\nu )\) may depend also on \(\nu \).
We wish to construct a trigonometric polynomial \(L_n\), of degree \(\beta (n-1)\), which satisfies analogues of (1.3) through (1.5), not only for \(h\approx \frac{\pi }{n}\), but rather, for any \(\frac{\pi }{n}\le h\le \mathop {\text {const}}\nolimits \).
Theorem 1.1
For each \(m\in \mathbb N\), \(\beta \in \mathbb N\) and \(0<\epsilon \le 1\), there are constants \(K_1>0\) and \(K_2\), depending only on m, \(\beta \) and \(\epsilon \), such that if \(\frac{\pi }{n}\le h\le \frac{\pi }{m}\), then the trigonometric polynomial
of degree \(\le \beta (n-1)\), satisfies
and for all \(0\le \nu \le m-1\),
and
Remark 1.2
The above constants may be replaced by \(K_1:=C_1\epsilon ^{m-1}\) and \(K_2:=C_2\epsilon ^{-2\beta }\), where \(C_1\) and \(C_2\) depend only on m and \(\beta \).
We apply Theorem 1.1 to obtain an interpolation result. Namely,
Theorem 1.3
Given \(n,s,\eta \in \mathbb N\) and \(0<\epsilon \le 1/2\). For \(\frac{\pi }{n}\le h\le \frac{\pi }{s+2}\), denote
Given a collection \(\{z_i\}_{i=1}^{2s}\), of distinct points in \([-\pi ,\pi )\). Let l, \(0\le l\le 2s\), be such that
and
Assume that f is defined in \(\dot{O}\), and if \(l\ge 1\), assume that \(f\in C^{l-1}(\dot{O})\) and satisfies,
Then, there exists a constant \(c=c(s,\eta ,\epsilon )\) and a trigonometric polynomial \(D_l\) of degree \(\le ([\eta /2]+2s+1)n\), such that
and its derivative \(d_l:=D'_l\) satisfies,
and, for all \(\nu =0,\dots ,2s\),
where
Finally, in Sect. 6, we apply Theorem 1.3 to obtain a trigonometric polynomial which approximates a \(2\pi \)-periodic continuous piecewise algebraic polynomial, and the derivative of which, interpolates the derivative of the latter at a given collection of points (obviously, not knots).
Throughout the paper we will have positive constants c and C that may differ from one another on different occurrences even if they appear in the same line.
2 Pointwise Bernstein Inequality
We extend the well known Bernstein inequality
which is valid for all trigonometric polynomials \(T_n\), of degree \(\le n\), and for all \(r\in \mathbb N\) and \(\nu \in \mathbb N\), into a pointwise version. Namely,
Lemma 2.1
For arbitrary trigonometric polynomial \(T_n\) of degree \(\le n\), any \(r\in \mathbb N\) and each natural \(\nu \le r\) the inequality
holds.
Proof
Without loss of generality assume that \(\Vert T_n\Vert =1\), so we have to prove the inequality
Then Bernstein inequality implies \(\Vert T_n^{(\nu )}\Vert \le n^\nu \) and
Assuming by induction, that for some \(\nu -1<r\) the inequality (2.1) holds for all \(r\in \mathbb N\), we get
which is (2.2). \(\square \)
Applying Lemma 2.1 to the polynomial \(T_n(u)=\frac{\sin {nu}}{n\sin u}\), we readily obtain (1.5).
3 Fast Decreasing Trigonometric Polynomials
Proof of Theorem 1.1
Since J is a trigonometric polynomial of degree \(<\beta n\), \(L_n\) is also a trigonometric polynomial of degree \(<\beta n\).
First, we have
and (1.6) is proved.
Evidently, for every \(\nu =1,\dots , m-1\), there is a \(\theta =\theta _\nu \in [x-\nu h/2,x+\nu h/2]\), such that
Now, for any \(a\in \mathbb R\),
Thus, (3.1) implies,
which is (1.8).
In order to prove (1.9), take \(\frac{(m+\epsilon )h}{2}\le |x|\le \pi \) and \(|\theta -x|\le \nu h/2\). If
then
which implies,
Here and in the rest of the proof C and \(C^*\) depend only on \(\beta \).
If, on the other hand, \(|\Theta |>\pi \), then \(\pi <|\Theta |\le \frac{3\pi }{2}\), which implies
so that (3.2) is valid in this case too.
Combined, (3.1) and (3.2) yield
where, for \(|x|\ge \frac{(m+\epsilon )h}{2}\), we applied the inequalities \(|x|-\frac{mh}{2}\ge \frac{\epsilon |x|}{m+1}\) and \(|x|-\frac{mh}{2}\ge \frac{\epsilon h}{2}\). Thus, we obtain (1.9).
In order to prove (1.7), we observe that if \(|a|\le h/2\), then
For \(m=1\), this readily yields (1.7). Thus, let \(m>1\) and denote
Note that if \(t_j\in H\), \(1\le j\le m-1\), then \(-h/2\le a:=x+t_1+\cdots +t_{m-1}\le h/2\).
Hence, for \(|x|\le \frac{(m-1)h}{2}\),
where we used the fact that \(|H|\ge \frac{h}{2(m-1)}\).
Finally, if \(\frac{(m-1)h}{2}<|x|\le \frac{(m-\epsilon )h}{2}\), then
Substituting in (3.4), completes the proof. \(\square \)
4 Interpolating Trigonometric Polynomials
Proof of Theorem 1.3
If \(l=0\), then (1.14) is empty, so we may take \(D_0(x)\equiv 0\).
We proceed by induction. By the induction assumption, there is a polynomial \(D_{l-1}\), \(1\le l\le 2s\), satisfying (1.14) through (1.16) with \(l-1\) instead of l, and with any \(\tilde{z}_l\in \ddot{O}\), \(\tilde{z}_l\notin \{z_i\}_{i=l+1}^{2s}\), instead of the given \(z_l\in \dot{O}\).
We will construct the derivative \(d_l\) and then put \(D_l(x):=\int _{-\pi }^x d_l(t)\,dt\).
To construct \(d_l\) we first note that for the polynomial \(L_n\), defined by Theorem 1.1 with \(\beta =[\eta /2]+2s+1\) and \(m=2s+1\), we have for \(x\in [-\pi ,\pi ]\),
Here and in the rest of the proof \(c=c(s,\eta ,\epsilon )\).
We will show that the desired polynomial \(d_l\) may be taken in the form
where
If \(l=1\), we mean \(\prod _{q=1}^0=1\), and recall that \(d_{l-1}=d_0\equiv 0\). Similarly, \(\prod _{q=2s+1}^{2s}=1\).
Evidently, \(d_l\) is a trigonometric polynomial of degree \(\le \beta (n-1)+2s\), and (1.14) and (1.15) hold.
We first estimate the polynomials \(\hat{d}_l\) and \(\breve{d}_l\), and their derivatives.
By the induction assumption, (1.12) and (1.16) imply,
while (1.14) yields,
Thus, we have
where the middle term is the divided difference of \(f-d_{l-1}\), and in the last inequality we used the fact that \(\theta \in \dot{O}\).
In order to estimate \(\hat{I}(x)\) and \(\breve{I}(x)\), and their derivatives, we observe that
and
Also
Hence, for \(x\in I\) and \(z\in \dot{O}\), we have
and for \(x\in I\) and \(z\in I\setminus \ddot{O}\), we have
Therefore, we write \(\hat{I}(x)=h^{l-1}\prod _{q=1}^{2s}\alpha _q(x)\), \(x\in I\), where for each \(1\le q\le 2s\),
This, in turn, yields for each \(0\le \nu \le 2s\),
Combining with (4.2), (1.7) and (4.1), we obtain for each \(0\le \nu \le 2s\),
Similarly,
whence,
It follows by (1.13) with \(l-1\), that
By virtue of (4.3) and (4.4), we obtain
and, in particular,
So, in order to complete the proof of (1.13) and (1.16), we will prove that
To this end, we note that if \(x\in [-sh,sh]\subset \dot{O}\), then
and
Hence,
Now, the algebraic polynomial,
of degree \(2l+1\), satisfies, by Markov’s inequality,
and (4.5) is proved. Thus, the proofs of (1.13) and (1.16) are complete. \(\square \)
5 An Auxiliary Lemma
For \(j\in \mathbb Z\), let
Denote by \(\mathbb P_k\), the space of algebraic polynomials of degree \(<k\), and by \(\widetilde{\Sigma }_{k,n}\), the space of \(2\pi \)-periodic continuous piecewise algebraic polynomials S, of degree \(<k\), with knots \(x_j\), that is,
Given \(S\in \widetilde{\Sigma }_{k,n}\), \(n_1, \beta \in \mathbb N\) and \(J_{n_1}:=J_{n_1,\beta }\), let \(\nu \in \mathbb N_0\) and denote
Lemma 5.1
Let \(S\in \widetilde{\Sigma }_{k,n}\)and let \(\sigma \in \mathbb N\). For each \(\nu \in \mathbb N\) we have
Proof
We first prove that for each \(\nu \in \mathbb N\),
To this end, we observe that since S is differentiable of any degree \(l\in \mathbb N\), except at a final number of points in any compact interval, the following integrals exist, for all \(l\in \mathbb N_0\), and are equal.
where for the second equation we used the fact that the integrand is \(2\pi \sigma \)-periodic.
Now,
Hence,
where for the last equation we used the fact that \(p^{(l)}_{-n\sigma }(x_{-n\sigma })=p^{(l)}_{n\sigma }(x_{n\sigma })\).
Thus (5.3) is proved.
Since \(B_{0,n_1}(x)\equiv 0\), the lemma follows by induction. \(\square \)
Remark 5.2
Since S is a continuous function, the \((l=1)\)-term of (5.2) vanishes, so that the sum begins with \(l=2\).
6 Approximating a Piecewise Polynomial
For \(f\in C[a,b]\), let
and, for \(k>1\), let
Denote by
the kth modulus of smoothness of f. (See [1, Chapter 2, Section 7] for properties of moduli of smoothness.)
Similarly, for \(f\in \widetilde{C}\), the space of \(2\pi \)-periodic functions on \(\mathbb R\), let \(\Delta _h(f,x)=\Delta ^1_h(f,x):=f(x+h)-f(x)\), and for \(k>1\), let \(\Delta ^k_h(f,x):=\Delta _h\left( \Delta ^{k-1}_h(f,\cdot ),x\right) \).
Finally, denote by
the kth modulus of smoothness of f. Note that for such f, \(\omega _k(f,t)=\omega _k(f,t;[-2\pi ,2\pi ])\) for \(0<t<2\pi /k\).
We call a closed interval E a proper interval, if \(E=[x_{j_*},x_{j^*}]\) for some indices \(j_*\) and \(j^*\), and \(x_{j^*}-x_{j_*}<2\pi \).
Let \(Y_s:=\{y_i\}_{i\in \mathbb Z}\), \(s\ge 1\), be a set of points, such that \(y_i<y_{i+1}\) and \(y_{i+2s}=y_i+2\pi ,\quad i\in \mathbb Z\).
For each \(i\in \mathbb Z\), let \(j_i\) be the index such that \(y_i\in [x_{j_i},x_{j_i+1})\). We denote by O the interior of the union
We will write \(S\in \widetilde{\Sigma }_{k,n}(Y_s)\), if \(S\in \widetilde{\Sigma }_{k,n}\) and \(p_{j\pm 1}\equiv p_j\) for \(x_j\in O\).
For \(x\in E\) and \(\eta \in \mathbb N\), denote
and, finally, let
We devote this section to proving,
Theorem 6.1
Let \(n_1\ge n\) and \(S\in \widetilde{\Sigma }_{k,n}(Y_s)\). Then there is a trigonometric polynomial \(\mathcal T\) of degree \(<cn_1\), such that
and if E is a proper interval, then
Here and in the sequel c and C denote constants which depend only on some or all the parameters k, s and \(\eta \).
Let
be the connected components of the set O, enumerated from right to left, and set
We need a few lemmas.
Lemma 6.2
Let \(n_1\ge n\), \(1\le \nu \le 2s+1\), \(1\le \sigma \le k\), \(S\in \tilde{\Sigma }_{k,n}(Y_s)\) \(\eta \ge 2s\) and \(J_{n_1}:=J_{n_1,\eta }\). If E is a proper interval and \(O_\mu \subset E\), then for all \(x\in \tilde{O}_\mu \),
Proof
Clearly, for any \(j^0\in \mathbb Z\), we may rewrite (5.2) as
where we use the fact that S is continuous and \(2\pi \)-periodic, so that \(p_j(x_j)=p_{j-1}(x_j)\), \(j\in \mathbb Z\), and \(p_j\equiv p_{j-1}\) for all j such that \(x_j\in O\), and that \(J_{n_1}(t/\sigma )\) is \(2\pi \sigma \)-periodic.
In particular we may take \(j^0\) where \(x\in I_{j^0}\). Thus, without loss of generality, we may assume that \(x\in I_0\) and \(j^0=0\). Then, for each j, \(-n\sigma +1\le j\le n\sigma \), we have \(\frac{|x-x_j|}{\sigma }\le \pi \).
By virtue of (1.5) we obtain for each \(\hat{j}\), \(-n\sigma +1\le \hat{j}\le \mu ^-\),
Similarly, (1.5) implies for each \(\breve{j}\), \(\mu ^+\le \breve{j}\le n\sigma \),
Let \(E=[x_{j_*},x_{j^*}]\). By Markov’s inequality \(|p^{(l)}_j(x_j)-p^{(l)}_{j-1}(x_j)|\le cn^l\omega _k(S,1/n;E)\), if either \(j_*<j\le \mu ^-\), or \(\mu ^+\le j<j^*\). Also \(|p^{(l)}_j(x_j)-p^{(l)}_{j-1}(x_j)|\le cn^l\omega _k(S,1/n)\) for all \(j\in \mathbb Z\).
We have to separate the proof for \(\sigma =1\) and \(\sigma \ge 2\). We begin with the latter, so that
Hence,
and
Similarly,
If \(\sigma =1\), then we may have \(-n+1\le j_*<j^*\le n\), and the proof follows verbatim as above. Otherwise, we may have that \(j_*\le -n\), so that (6.8) is irrelevant, while in the summation in (6.9) we replace \(j=-j_*+1\) by \(j=-n+1\), where we note that \(\mu ^->-n\); or we may have \(j^*>n\), so that we replace the upper end of the summation \(j^*\) with n, where we note that \(\mu ^+<n\). This completes the proof. \(\square \)
Lemma 6.3
Let \(n_1\ge n\), \(1\le \nu \le 2s+1\), \(S\in \tilde{\Sigma }_{k,n}(Y_s)\) \(\eta \ge 2s\) and \(J_{n_1}:=J_{n_1,\eta }\), and let
be the trigonometric polynomial of degree \(<\eta n_1\). Then
If E is a proper interval, then
where A(x, E) was defined in (6.1).
Moreover, if \(O_\mu \subset E\), then for all \(x\in \tilde{O}_\mu \) and \(1\le \nu \le 2s+1\),
Proof
The inequality \(\Vert S-T\Vert \le c\omega _k(S,1/n_1)\le c\omega _k(S,1/n)\) is well-known.
Since S is a piecewise algebraic polynomial, it possesses all left- and right-hand derivatives at each \(x_j\), \(j\in \mathbb Z\). Thus, if it happens that \(x+\nu t=x_j\) for some \(j\in \mathbb Z\), then \(\Delta _t^k(S^{(\nu )},x)\) may not be well defined. But, for each fixed x, this may happen only for finitely many values of t, and would not influence the integration below. However, when we wish to estimate \(\Delta _t^k(S^{(\nu )},x)\) we must consider the collection of all (finitely many) possible values we may have by assigning the various appropriate left- or right-hand values that may occur.
Recall that in [2, Lemma 5.5], it was proved for a proper interval E, that if \(x,x+kt\in E\) and \((x+\ell t)\notin \{x_j\}_{j=-\infty }^\infty \), \(0\le \ell \le k\), then
Closely observing the proof of [2, Lemma 5.5], we see that (6.13) is valid also with our above relaxation. In addition, the restriction on the length of E, is irrelevant for the next inequality. Thus,we obtain,
If \(x\in E\), and \(d:=\text {dist}(x,\mathbb R\setminus E)\ge \pi /(2n)\), then
In particular, this is the case when \(x\in \tilde{O}_\mu \).
Similarly, if \(0<d<\pi /(2n)\), then
Thus, (6.11) is proved.
Finally, if \(x\in \tilde{O}_\mu \subset E\), then by (6.15) and (6.7),
and (6.12) follows. \(\square \)
If a proper interval E is such that its endpoints are not in O, we will call it a \(Y_s\)-proper interval.
For each \(\mu \in \mathbb Z\), let \(x_{\mu ^\circ }:=\frac{1}{2}(x_{\mu ^-}+x_{\mu ^-})\) be the midpoint of \(O_\mu =(x_{\mu ^-},x_{\mu ^-})\), and for each \(Y_s\)-proper interval, such that \(O_\mu \subset E\), let
Since \(\mathop {\text {dist}}\nolimits (x_{\mu ^\circ },\mathbb R\setminus E)\le C\mathop {\text {dist}}\nolimits (x,\mathbb R\setminus E)\), for all \(x\in \tilde{O}_\mu \), and \(\mathop {\text {dist}}\nolimits (x,\mathbb R\setminus E)\le C\mathop {\text {dist}}\nolimits (x_{\mu ^\circ },\mathbb R\setminus E)\), for all \(x\in O_\mu \), it follows that
and
Define
Finally, denote \(J_\mu :=[x_{\mu ^\circ }-\pi ,x_{\mu ^\circ }+\pi ]\), and let \(M_\mu \) be the \(2\pi \)-periodic function, defined on \(J_\mu \) by
Lemma 6.4
Let \(\mu \in \mathbb Z\) and \(n_1\ge n\). Then, for every \(Y_s\)-proper interval E and each \(x\in E\), we have
Proof
It is sufficient to prove (6.19) for \(x\in J_\mu \), and we let a \(Y_s\)-proper interval E be such that \(x\in E\).
First, assume that \(O_\mu \nsubseteq E\). Thus, there is an endpoint of E, say \(\gamma \), lying between x and \(x_{\mu ^\circ }\). Then \(\mathop {\text {dist}}\nolimits (x,\mathbb R\setminus E)\le |x-\gamma |\le |x-x_{\mu ^\circ }|\).
Hence,
Otherwise, \(O_\mu \subset E\).
If \(x\in O_\mu \), then (6.19) is trivial, since \(\Vert M_\mu \Vert =1\), and by (6.17)
Similarly, if \(x\in E\setminus O_\mu \) and \(|x-\gamma |\le |x_{\mu ^\circ }-\gamma |\), where now \(\gamma \) is the endpoint of E, closest to \(x_{\mu ^\circ }\), then
that yields (6.19).
Finally, if \(x\in E\setminus O_\mu \) and \(|x-\gamma |>|x_{\mu ^\circ }-\gamma |\), then assume, without loss of generality, that \(x_{\mu ^\circ }<\gamma \). Then, it follows that \(x+3\pi /(2n)\le x_{\mu ^\circ }\le \gamma -3\pi /(2n)\). Since \(g(u):=\frac{\gamma -u}{x_{\mu ^\circ }-u}\) is an increasing function for \(u<x_{\mu ^\circ }\), we have,
Hence,
Therefore, \(A_{\mu }M_\mu ^\eta (x)\le A(x,E)\). \(\square \)
We are ready to prove Theorem 6.1. It is easy to show that if an endpoint of a proper interval E belongs to O, say, to its connected component \(O_\mu \), then \(\omega _k(S,1/n;\overline{E\cup O_\mu })\le c\omega _k(S,1/n;E)\), whence \(A(x,\overline{E\cup O_\mu })\le CA(x,E)\), \(x\in E\). Therefore, we prove Theorem 6.1 for \(Y_s\)-proper intervals E and, without loss of generality, we assume that \(\eta \ge 2s\).
Proof of Theorem 6.1
We apply Theorem 1.3 for each fixed \(\mu \in \mathbb Z\), with \(\epsilon =1/12\), and \(n_1\) instead of n and h such that,
Thus,
Since \(\frac{3\pi }{n}\le |O_\mu |\le \frac{6s\pi }{n}\), we conclude that
and
Hence,
Note that all points \(y_i\in J_\mu \) lie either in \(\dot{O}_\mu \) or outside \(\ddot{O}_\mu \). Let l be the number of points \(y_i\in \dot{O}_\mu \).
Let T be the polynomial, guaranteed by Lemma 6.3, and denote
so that, for all \(0\le \nu \le l-1\), we have
Thus, f satisfies (1.12). Hence, (1.14) through (1.17), imply the existence of a polynomial \(d_l\), of degree \(<cn_1\), such that
and for all \(0\le \nu \le 2s\),
By (6.12), \(R_\mu \le ch^{-1}A_\mu \). Therefore, the polynomial
satisfies
and for all \(1\le \nu \le 2s+1\),
where in the last inequality we used the fact that \(\dot{O}_\mu \subset \tilde{O}_\mu \).
Finally, Lemma 6.4 implies that for every \(Y_s\)-proper interval E, we have
We will prove that the desired polynomial \(\mathcal T\) may be taken in the form
Indeed, (6.3) readily follows by (6.10) and (6.20).
We observe that,
where \(\pi (t)\) was defined in (6.2), and combined with (6.11) and (6.21) with \(\nu =1\), we obtain (6.4) for \(x\in E\setminus \tilde{O}\).
On the other hand, if \(x\in \tilde{O}_{\mu ^*}\subset E\), for some \(\mu ^*\in \mathbb Z\), then let \(y_{i_\ell }\in O_{\mu ^*}\), \(1\le \ell \le l\), and note that \(y_{i_\ell }\in \tilde{O}_{\mu ^*}\). Evidently, \(S'(y_{i_\ell })=\mathcal T'(y_{i_\ell })\), \(1\le \ell \le l\).
Applying (6.12) and (6.21) for all \(\mu \), all with \(\nu =l+1\), we obtain
Hence, for \(x\in \tilde{O}_{\mu ^*}\),
Thus, by (6.17), we conclude that
This completes the proof. \(\square \)
References
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Leviatan, D., Motorna, O.V. & Shevchuk, I.A. Fast Decreasing Trigonometric Polynomials and Applications. J Fourier Anal Appl 30, 28 (2024). https://doi.org/10.1007/s00041-024-10080-4
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DOI: https://doi.org/10.1007/s00041-024-10080-4
Keywords
- Fast decreasing trigonometric polynomials
- Interpolation by trigonometric polynomials
- Jackson-type estimates.