Notes on Bernstein spaces, sampling, Boas interpolation formulas and their extensions to Banach spaces

TL;DR

This survey reviews classical harmonic analysis results like Bernstein spaces, sampling, and interpolation formulas, and discusses their recent extensions to Banach spaces with bounded operator groups.

Contribution

It provides a comprehensive overview of extending harmonic analysis concepts to Banach spaces with bounded operator groups, highlighting recent developments.

Findings

Summarizes key results in Bernstein spaces and sampling theory.

Details extensions of interpolation formulas to Banach spaces.

Connects classical harmonic analysis with modern Banach space theory.

Abstract

This paper is essentially a survey on several classical results of harmonic analysis and their recent extensions to Banach spaces. The first part of the paper is a summary of some important results in such topics as Bernstein spaces, Shannon-type sampling, Riesz and Boas interpolation formulas. The second part contains extensions of these ideas to Banach spaces equipped with one-parameter uniformly bounded group of operators of class $C_{0}$.