On approximation of continuous functions by entire functions on subsets of the real line

TL;DR

This paper extends Bernstein's classical theorem to describe how continuous functions can be approximated by entire functions of exponential type on unbounded subsets of the real line, broadening understanding of function approximation.

Contribution

It generalizes Bernstein's theorem to unbounded subsets of the real line, providing new insights into approximation by entire functions of exponential type.

Findings

Approximation of continuous functions by entire functions of exponential type on unbounded sets.

Extension of classical Bernstein theorem to broader subsets of the real line.

New constructive descriptions for function classes on unbounded domains.

Abstract

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studded.