Sharp Lp-error estimates for sampling operators

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2023-10-01 DOI:10.1016/j.jat.2023.105941

Yurii Kolomoitsev , Tetiana Lomako

{"title":"Sharp Lp-error estimates for sampling operators","authors":"Yurii Kolomoitsev , Tetiana Lomako","doi":"10.1016/j.jat.2023.105941","DOIUrl":null,"url":null,"abstract":"<div>We study approximation properties of linear sampling operators in the spaces <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math> for <math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math>. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math>, find the exact order of decay of the corresponding <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-errors for particular classes of functions, and introduce a special <math><mi>K</mi></math>-functional and its realization suitable for studying smoothness properties of sampling operators.</div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Approximation Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021904523000795","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We study approximation properties of linear sampling operators in the spaces $L_{p}$ for $1 \leq p < \infty$ . By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in $L_{p}$ and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in $L_{p}$ , find the exact order of decay of the corresponding $L_{p}$ -errors for particular classes of functions, and introduce a special $K$ -functional and its realization suitable for studying smoothness properties of sampling operators.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

采样算子的夏普Lp误差估计

研究了空间Lp中线性采样算子对1≤p<；∞的逼近性质。通过Steklov平均，我们引入了一种新的光滑度度量，该度量同时包含关于Lp中函数的光滑度的信息和关于函数在采样点的行为的离散信息。新的光滑度度量使我们能够改进近似理论的几个经典结果，并将其扩展到线性采样算子的情况。特别地，我们得到了Lp中采样算子的直接和逆近似不等式，找到了特定函数类的相应Lp误差的精确衰减阶，并介绍了一种适用于研究采样算子光滑性的特殊K函数及其实现。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.

期刊最新文献

Optimization-aided construction of multivariate Chebyshev polynomials In search of a higher Bochner theorem Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials Editorial Board On the representability of a continuous multivariate function by sums of ridge functions