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Basic Relations Valid for the Bernstein Space $B^{p}_{\sigma}$ and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from $B^{p}_{\sigma}$

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Basic Relations Valid for the Bernstein Space $B^{p}_{\sigma}$ and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from $B^{p}_{\sigma}$

Content Provider	Springer Nature Link
Author	Stens, Rudolf L. Butzer, Paul L. Schmeisser, Gerhard
Copyright Year	2013
Abstract	There are various basic relations (equations and inequalities) that hold in Bernstein spaces $B_{\sigma}^{p}$ but are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, one may expect that the corresponding relation is not violated drastically. It should hold with a remainder that involves the distance of f from $B_{\sigma}^{p}$ .First we establish a hierarchy of spaces that generalize the Bernstein spaces and are suitable for our studies. It includes Hardy spaces, Sobolev spaces, Lipschitz classes and Fourier inversion classes. Next we introduce an appropriate metric for describing the distance of a function belonging to such a space from a Bernstein space. It will be used for estimating remainders and studying rates of convergence.In the main part, we present the desired extensions. Our considerations include the classical sampling formula by Whittaker-Kotel’nikov-Shannon, the sampling formula of Valiron-Tschakaloff, the differentiation formula of Boas, the reproducing kernel formula, the general Parseval formula, Bernstein’s inequality for the derivative and Nikol’skiĭ’s inequality estimating the $l^{p}(\mathbb{Z})$ norm in terms of the $L^{p}(\mathbb{R})$ norm. All the remainders are represented in terms of the Fourier transform of f and estimated from above by the new metric. Finally we show that the remainders can be continued to spaces where a Fourier transform need not exist and can be estimated in terms of the regularity of f.
Ending Page	375
Page Count	43
Starting Page	333
File Format	PDF
ISSN	10695869
e-ISSN	15315851
Journal	Journal of Fourier Analysis and Applications
Issue Number	2
Volume Number	19
Language	English
Publisher	SP Birkhäuser Verlag Boston
Publisher Date	2013-03-16
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Reproducing kernel formula Inequalities in approximation (Bernstein, Jackson, Nikol'skiÄ-type inequalities) Non-bandlimited functions Sampling formulae Signal, Image and Speech Processing Hardy spaces Formulae with remainders General Parseval formula Abstract Harmonic Analysis Banach spaces of continuous, differentiable or analytic functions Mathematical Methods in Physics Sampling theory Fourier Analysis Differentiation formulae Nikolskiĭ’s inequality Approximations and Expansions Bernstein’s inequality Remainders in approximation formulas Derivative-free error estimates Numerical differentiation Partial Differential Equations Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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Basic Relations Valid for the Bernstein Space $B^{p}_{\sigma}$ and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from $B^{p}_{\sigma}$