On weighted polynomial approximation

TL;DR

This paper proves that polynomials are dense in certain weighted continuous function spaces on rica, under conditions involving the growth of the weight function .

Contribution

It establishes the density of polynomials in weighted spaces with weights growing faster than any exponential, extending approximation theory.

Findings

Polynomials are dense in weighted spaces with specific growth conditions.

The weight function  must satisfy a super-exponential growth condition.

The result generalizes classical polynomial approximation results to new weighted contexts.

Abstract

Let $\varPhi:{\mathbb R}^n \to [1, \infty)$ be a semi-continuous from below function such that $\lim \limits_{x \to \infty} \displaystyle \frac {\ln \varPhi(x)} {\Vert x \Vert} = +\infty$. It is shown that polynomials are dense in $C_{\varPhi}({\mathbb R}^n)$.