Polynomial selections and separation by polynomials

TL;DR

This paper establishes conditions for when two real functions can be separated by a polynomial, with applications to convex functions and stability results, advancing understanding of polynomial approximation and separation.

Contribution

It provides necessary and sufficient conditions for polynomial separation of functions, including convex functions and stability implications.

Findings

Conditions for polynomial separation of functions are characterized.

Existence of polynomial separation for convex functions of higher order.

Application to Hyers-Ulam stability results.

Abstract

Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex functions of higher order. Another application is some Hyers-Ulam-stability-type result.