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Mathematics
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Advances in Approximation Theory and Numerical Functional Analysis
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Advances in Approximation Theory and Numerical Functional Analysis

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

Deadline for manuscript submissions: 31 March 2025 | Viewed by 2562

Special Issue Editor

Prof. Dr. Gancho Tachev

E-Mail Website
Guest Editor
Department of Mathematics, University of Architecture Civil Engineering and Geodesy, 1046 Sofia, Bulgaria
Interests: approximation by linear positive operators; Bernstein polynomials; inequalities for polynomials; quantitative Voronovskaja-type estimates

Special Issue Information

Dear Colleagues,

The purpose of this Special Issue is to present recent results in the fields of mathematical analysis, approximation theory, and numerical functional analysis, as well as to describe the numerical methods currently being used in important areas of applications of approximation theory.

Some classics of approximation theory include quantitative direct and inverse estimates for approximation by positive linear operators (p.l.o.), Voronovskaja-type theorems, weighted approximation, linking-type operators, simultaneous approximation by some classical operators, best constants in classical and new polynomial inequalities and many others are the focus of this book. Modifications of some p.l.o. preserving certain exponential functions have also been intensively studied in recent decades. The error of approximation is measured by the usual moduli of smoothness, as well as by appropriate K-functionals. Another important branch of approximation by p.l.o. is to use their linear combinations and in this way to increase the rate of approximation. Different types of inequalities for algebraic and trigonometric polynomials like inequalities of Bernstein, Markov and best constants in various estimates are also attractive topics of approximation theory. Splines, including Schoenberg variation-diminishing spline operators and their numerous applications in numerical analysis, are also an interesting branch in approximation theory. Interpolation by these modern tools in one or more dimensions have been studied intensively in recent decades.

Prof. Dr. Gancho Tachev
Guest Editor

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

  • Bernstein polynomials
  • linear combinations
  • moduli of smoothness
  • rate of approximation

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Published Papers (2 papers)

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Research

21 pages, 321 KiB  
Article
Approximation by Schurer Type λ-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties
by Abdullah Alotaibi
Mathematics 2024, 12(21), 3310; https://doi.org/10.3390/math12213310 - 22 Oct 2024
Viewed by 666
Abstract
In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used [...] Read more.
In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained. Full article
(This article belongs to the Special Issue Advances in Approximation Theory and Numerical Functional Analysis)
23 pages, 330 KiB  
Article
Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm
by Xu Xu, Jinyu Guo, Peixin Ye and Wenhui Zhang
Mathematics 2023, 11(9), 2020; https://doi.org/10.3390/math11092020 - 24 Apr 2023
Cited by 1 | Viewed by 1225
Abstract
We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on [...] Read more.
We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A1(D) and the closure of the convex hull of the dictionary D is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on A1(D) is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm. Full article
(This article belongs to the Special Issue Advances in Approximation Theory and Numerical Functional Analysis)
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