On incompleteness of polynomials in some weighted spaces on half line

TL;DR

This paper investigates the conditions under which polynomials are complete in weighted Lp-spaces on the half line, revealing that they are not complete for many common weight functions including certain exponential weights.

Contribution

It demonstrates the incompleteness of polynomials in weighted Lp-spaces for a broad class of weights, extending understanding of polynomial approximation in these spaces.

Findings

Polynomials are not complete in weighted Lp-spaces for weights like exp(- r t^q) with r>0 and q in (0,1)

Completeness fails for a wide class of weights including exponential types

Provides conditions under which polynomial completeness does not hold

Abstract

The paper studies completeness of the polynomials in weighted $L_p$-spaces on half line. It is shown that the completeness of polynomials does not hold for a wide class of weights, including the weights $\exp(- r t^q)$ with $r>0$ and $q\in (0,1)$.