Abstract
The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane.
1 Introduction
Let ℝd be the Euclidean space of vectors . Let be a function 2π-periodic in each variable xi, and summable on the set 𝕋d = [−π; π]d, i.e., f ∈ L(𝕋d), let
be the complete Fourier series of function f, where
are the Fourier coefficients of the function f, corresponding to the vectors , and is the number of zero coordinates of the vector .
Let be the fixed set of infinite triangular matrices of numbers such that , , ki ≥ ni. Denote , and . If , then . For function f ∈ L(𝕋d) the set defines a family of trigonometric polynomials
The polynomials are called rectangular linear means for In particular, if then are the rectangular partial sums of , and if , then
are the rectangular Fejér means of .
Basic results relating to the approximation of functional classes by linear methods of summation of Fourier series can be found in books Timan [1], Lorentz [2], and Dyachenko [3]. Linear summation methods are widely used both for the solution of practical problems and for development of more advanced approximation methods. This chapter of approximation theory has been intensively developed over the past decades [4–9]. Here it is difficult to mention all the relevant published research papers in this area. Recently, we have seen the publication of several important works [10–15].
Let C(𝕋d) be the space of continuous 2π-periodic in each variable's functions with the norm
Let be the arbitrary subset of the set where r is the number of elements of the set . Denote by , the set of functions f ∈ C(𝕋d) such that , the series
are the Fourier series of certain functions , which are almost everywhere bounded by a unity, and the Fourier series of functions do not contain terms independent of the variables xi, .
For example, in the case d = 2, the series (Equation 1) is as follows:
In the one-dimensional case, the classes Cq(𝕋1), q∈(0;1) consist of continuous 2π-periodic functions, given by the convolution
where
is the well-known Poisson kernel, the function satisfies almost everywhere the conditions ,
In this work, we consider the problem of the exact upper bound for the approximation of periodic functions by linear means of the Fourier series. We employed methods for studying integral representations of deviations of polynomials, generated by linear summation methods of Fourier series of continuous periodic functions, developed in the works of Nikolskii [16], Telyakovskii [17], Stepanets [18], and others. This topic is currently being developed in the works of many authors [19–21].
Nikolskii [22] established the asymptotic equality as n → ∞
where is the complete elliptic integral of the first kind and O(1) is a quantity uniformly bounded with respect to n. Regarding the summability of Fourier series by Fejér means σn[f], we proved the following two theorems [23–25].
Theorem 1. Letq0be the only root of the equationq4 − 2q3 − 2q2 − 2q+1 = 0, that belongs to the interval (0;1), Ifq ∈ (0;q0], then the equality hold asn → ∞
whereO(1) is a quantity uniformly bounded with respect ton.
Theorem 2. Ifq ∈ [q0; 1), then the equality hold asn → ∞
whereO(1) is uniformly bounded with respect ton, q.
The purpose of this paper is to present the asymptotic equalities for upper bounds of deviations of rectangular Fejér means taken over multidimensional analogs of classes Cq(𝕋1). Similar asymptotic expansions for other rectangular linear methods can be found in Rukasov et al. [26] and Rovenska [27].
2 Result
The main result is the following.
Theorem 3. Let. Then
where
q0is the only root of the equationq4−2q3 − 2q2 − 2q+1 = 0, that belongs to the interval (0;1), q0 = 0.346…, O(1) is a quantity, uniformly bounded with respect toqi, ni, .
Proof
First we find the upper estimate for the quantity
Based on Theorem 1 in Rukasov et al. [26], , the equality holds
In Novikov et al. [24] and Rovenska [25] it was shown that
where
and tq is determined by the condition
Combining Equations 4, 5, and 6, we obtain
Next, we find the lower estimate of Equation 3. We construct the function for which estimate Equation 7 cannot be improved. Based on equality Equation 3 we have
Since the functions satisfy the condition almost everywhere, and
then
Denote by an arbitrary continuation on the set 𝕋d of the function , and denote by , the function, such that
Let It's clear that . Therefore, we have
Combining Equations 5, 7, and 8, we obtain equality (Equation 2). The proof is complete.
Remark 1. Formula Equation 2 is asymptotically exact for any
Remark 2. In the case d = 2, formula Equation 2 is simplified as follows:
3 Conclusion
In this study, we propose an approach to define the multidimensional analogs of classes of Poisson integrals, which allows us to take into account the rate of decrease of each sequence that determine the class. The problem connected with the search for upper bounds of approximation errors with respect to a fixed class of functions and with the choice of an approximation tool is considered.In the certain case, our approach turned out to be effective for obtaining exact asymptotic. The key point in this approach is to construct the function that implements the upper bound.
Our study may be useful for solving the upper bound problem in other particular cases. In particular, our ideas can be used to obtain the corresponding asymptotic equalities on classes, which in one-dimensional cases are determined by the Poisson kernels , β ∈ ℝ, etc.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.
Author contributions
OR: Writing – review & editing, Writing – original draft.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This research during 2020–2023 was supported by the Volkswagen Foundation project “From Modeling and Analysis to Approximation.”
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
linear method of approximation, extremal problem of approximation theory, Poisson integral, Fejér mean, exact asymptotic
Citation
Rovenska O (2024) Approximation of classes of Poisson integrals by rectangular Fejér means. Front. Appl. Math. Stat. 10:1437247. doi: 10.3389/fams.2024.1437247
Received
23 May 2024
Accepted
10 July 2024
Published
24 July 2024
Volume
10 - 2024
Edited by
Kateryna Buryachenko, Humboldt University of Berlin, Germany
Reviewed by
Tuncer Acar, Selçuk University, Türkiye
Bogdan Szal, University of Zielona Góra, Poland
Updates
Copyright
© 2024 Rovenska.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Olga Rovenska rovenskaya.olga.math@gmail.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.