Optimization properties of generalized Chebyshev-Poisson integrals
type: Item
instance of: scholarly article
title: Optimization properties of generalized Chebyshev-Poisson integrals
author: A. Yu. Mishchuk
author: Arsen M. Shutovskyi
published in: Cybernetics and Systems Analysis
publication date: 2024-09-25T00:00:00Z
review text: In this paper, an extremal problem of theory of approximating functions defined on a segment by linear methods of summing Fourier series is solved. Namely, it is raised the issue on the approximation of functions from the class \(H^1\) by generalized Chebyshev-Poisson integrals\N\[\N\begin{multlined}\NP_{s,q}(\rho;f;x)=\frac1\pi \int_{-1}^1\Big\{1+\sum_{k=1}^{+\infty}\rho^{k}[1+sk(1+\rho)(1-\rho)^{q}]\cos k(\arccos x-\arccos y)\\\N+\sum_{k=1}^{+\infty}\rho^{k}[1+sk(1+\rho)(1-\rho)^{q}]\cos k(\arccos x+\arccos y)\Big\}\frac{f(y)dy}{\sqrt{1-y^2}},\N\end{multlined}\tag{1}\N\]\Nwhere \(0\leq s\leq 1/2\) and \(q\geq 1\). \N\NIn particular, the authors established an exact equality for the quantity\N\[\N\mathcal{E}(H^1;P_{s,q}(\rho;x))=\sup_{f\in H^1} |f(x)-P_{s,q}(\rho;f;x)|.\tag{2}\N\]\NThe positive operator $(1)$ has a number of significant advantages in comparison with Abel-Poisson integrals and biharmonic Poisson integrals, which also belong to the class of positive operators. After all, the dependence of the Chebyshev-Poisson operator $(1)$ on two additional parameters \(s\) and \(q\) opens up wide opportunities for investigating the optimality of a solution to an extremal problem of the type $(2)$. \N\NThe results obtained in the present research summarize the results established in [\textit{J. I. Rusecki}, Sib. Mat. Zh. 9, 136--144 (1968; Zbl 0172.34102)] etc.
reviewed by: F. G. Abdullayev
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 42A99
zbMATH DE Number: 7918617
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: generalized Chebyshev-Poisson integral
zbMATH Keywords: class of Hölder functions
zbMATH Keywords: optimization problem
MaRDI profile type: Publication
cites work: Differential Games with Impulse Control
cites work: Game problems for fractional quasilinear systems.
cites work: On linear differential games with integral constraints
cites work: Finding solution subspaces of the Laplace and heat equations isometric to spaces of real functions, and some of their applications
cites work: Some asymptotic properties of the solutions of Laplace equations in a unit disk
cites work: Isometry of the subspaces of solutions of systems of differential equations to the spaces of real functions
cites work: Approximation of continuous functions given on the real axis by three-harmonic Poisson operators
cites work: On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric
cites work: Approximation of the classes \({W}_{\beta, \infty}^R\) by generalized Abel-Poisson integrals
cites work: Representation of Weierstrass integral via Poisson integrals
cites work: Approximation of classes \({C}_{\beta, \infty}^{\psi }\) by three-harmonic Poisson integrals in uniform metric (low smoothness)
cites work: Q4791971
cites work: Exact values of the approximations of differentiable functions by Poisson-type integrals
cites work: Approximation of (ψ, β)-Differentiable Functions Defined on the Real Axis by Abel-Poisson Operators
cites work: Some applied aspects of the Dirac delta function
cites work: On some asymptotic properties of solutions to biharmonic equations
cites work: Approximation of \((\psi, \beta)\)-differentiable functions of low smoothness by biharmonic Poisson integrals
cites work: Approximating properties of biharmonic Poisson operators in the classes \(\widehat{L}_{\beta,1}^\psi\)
cites work: Taylor series of biharmonic Poisson integral for upper half-plane
cites work: Oscillatory conflict-control processes
cites work: Approximation of functions from the class ${\ifmmode\expandafter\hat\else\expandafter\^\fi{C}}^\psi_{\beta,\infty}$ by Poisson biharmonic operators in the uniform metric
cites work: Approximation of functions from the classes \(C_{\beta, \infty}^{\psi}\) by biharmonic Poisson integrals
cites work: Approximation of functions defined on the real axis by operators generated by λ-methods of summation of their Fourier integrals
cites work: Q5558572
cites work: Approximation of conjugate differentiable functions by their Abel–Poisson integrals
cites work: Approximation properties of Abel-Poisson integrals on the classes of differentiable functions, defined by means of modulus of continuity
cites work: Asymptotic properties of the solutions of higher-order differential equations on generalized Hölder classes
cites work: Conditions of convergence almost everywhere for the convolution of a function with delta-shaped kernel to this function
cites work: The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions
cites work: Simple pursuit of one evader by a group
cites work: Fourier transform of the summatory Abel-Poisson function
cites work: Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions
cites work: On the asymptotics of some Weierstrass functions
cites work: Multivalued mappings and their selectors in the theory of conflict-controlled processes
cites work: A linear evasion problem for interacting groups of objects
DOI: 10.1007/S10559-024-00700-8
description: scientific article; zbMATH DE number 7918617