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Mathematics
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Fourier Analysis, Approximation Theory and Applications
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Fourier Analysis, Approximation Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (23 January 2024) | Viewed by 2872

Special Issue Editors

Prof. Dr. Kazaros Kazarian

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Sciences, C-XVII, Autonomous University of Madrid, 28049 Madrid, Spain
Interests: Fourier analysis; theory of orthogonal series; functional analysis; numerical analysis
Prof. Dr. Mikhail Dyachenko

E-Mail Website
Guest Editor
Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia; Lomonosov Moscow State University, Moscow 119992, Russia
Interests: Fourier analysis; theory of orthogonal series; approximation theory

Special Issue Information

Dear Colleagues,

Fourier analysis took root in Europe more than two hundred years ago, and since then, it has given rise to several new concepts and theories in different areas of contemporary mathematics. The first steps in the development of the theory were related to the representation of an arbitrary function by a trigonometric series. The concept of the Lebesgue integral strongly encourages the further development of the theory, considering functions from the different classes of functions which can be represented by series which converge or are summable in different senses.  With different types of applications, one can study series with respect to other types of orthogonal systems. To represent functions of various variables, multiple orthogonal series are considered.  Fourier analysis has many scientific applications, particularly in boundary value problems, approximation theory, signal processing, digital image processing and others.

In this Special Issue, we encourage submissions of up-to-date results related to classical approaches of Fourier analysis and its applications.

Prof. Dr. Kazaros Kazarian
Prof. Dr. Mikhail Dyachenko
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fourier series
  • orthogonal series, convergence and summation
  • approximation theory
  • representation of functions, bases in different senses
  • boundary value problems
  • signal processing

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Published Papers (3 papers)

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Research

19 pages, 358 KiB  
Article
Nearest Neighbor Recurrence Relations for Meixner–Angelesco Multiple Orthogonal Polynomials of the Second Kind
by Jorge Arvesú and Alejandro J. Quintero Roba
Mathematics 2024, 12(1), 62; https://doi.org/10.3390/math12010062 - 24 Dec 2023
Viewed by 681
Abstract
This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogs of Pascal distributions on a legged star-like set. The emphasis is placed on the algebraic properties, [...] Read more.
This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogs of Pascal distributions on a legged star-like set. The emphasis is placed on the algebraic properties, such as the raising operators, the Rodrigues-type formula, the explicit expression of the polynomials, and the nearest neighbor recurrence relations. Full article
(This article belongs to the Special Issue Fourier Analysis, Approximation Theory and Applications)
17 pages, 318 KiB  
Article
Representations by Beurling Systems
by Kazaros Kazarian
Mathematics 2023, 11(17), 3663; https://doi.org/10.3390/math11173663 - 24 Aug 2023
Viewed by 640
Abstract
We prove that a Beurling system with F∈Hp(D),1≤p<∞ is an M—basis in Hp(D) with an explicit dual system. Any function [...] Read more.
We prove that a Beurling system with F∈Hp(D),1≤p<∞ is an M—basis in Hp(D) with an explicit dual system. Any function f∈Hp(D),1≤p<∞ can be expanded as a series by the system {zmF(z)}m=0∞. For different summation methods, we characterize the outer functions F for which the expansion with respect to the corresponding Beurling system converges to f. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis. Full article
(This article belongs to the Special Issue Fourier Analysis, Approximation Theory and Applications)
15 pages, 312 KiB  
Article
On the Rate of Convergence of Greedy Algorithms
by Vladimir Temlyakov
Mathematics 2023, 11(11), 2559; https://doi.org/10.3390/math11112559 - 2 Jun 2023
Cited by 2 | Viewed by 1097
Abstract
In this paper, a new criterion for the evaluation of the theoretical efficiency of a greedy algorithm is suggested. Using this criterion, we prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of [...] Read more.
In this paper, a new criterion for the evaluation of the theoretical efficiency of a greedy algorithm is suggested. Using this criterion, we prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of Hilbert spaces and the more general case of Banach spaces. The new component of this paper is that we bound the error of approximation by the product of two norms—the norm of f and the A1-norm of f. Typically, only the A1-norm of f is used. In particular, we establish that some greedy algorithms (Pure Greedy Algorithm (PGA) and its modifications) are as good as the Orthogonal Greedy Algorithm (OGA) in this new sense of the rate of convergence, while it is known that the PGA is much worse than the OGA in the standard sense. Our new results provide better bounds for the accuracy than known results in the case of small ∥f∥. Full article
(This article belongs to the Special Issue Fourier Analysis, Approximation Theory and Applications)
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