On the density of polynomials under perturbations of the measure

TL;DR

This paper discusses the density of polynomials in weighted L2 spaces and presents a flawed proof claiming that polynomial density is preserved under certain measure perturbations supported on finite sets.

Contribution

It introduces a proof attempting to establish polynomial density preservation under finite measure perturbations, which is shown to be erroneous.

Findings

The proof claiming polynomial density preservation is incorrect.

Polynomial density in L2 spaces can be affected by measure perturbations.

The paper highlights the need for careful validation of measure perturbation effects.

Abstract

This paper presents an erroneous proof that if the polynomials are dense in $L_2(\mathbb{R}, \rho)$, then they are dense in $L_2(\mathbb{R}, \rho+\mu)$ where $\mu$ is a measure supported on a finite set of points.