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 $B^{p}_{\sigma}$

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There are various basic relations (equations and inequalities) that hold in Bernstein spaces $B_{\sigma}^{p}$ but are no longer valid in larger spaces.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.