Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic Versus Analytic Density of Polynomials

Brian Simanek, Richard Wellman

Abstract

We show that under very mild conditions on a measure $μ$ on the real line, the span of $\{x^n\}_{n=j}^{\infty}$ is dense in $L^2(μ)$ for any $j\in\mathbb{N}$. We also present a slightly weaker result with an interesting proof that uses Sobolev orthogonality.

Algebraic Versus Analytic Density of Polynomials

Abstract

We show that under very mild conditions on a measure on the real line, the span of is dense in for any . We also present a slightly weaker result with an interesting proof that uses Sobolev orthogonality.

Paper Structure

This paper contains 3 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. A First Approach
  3. A Second Approach

Key Result

Theorem \oldthetheorem

For any $j\in\mathbb{N}$, finite linear combinations of the polynomials $\{x^n\}_{n=j}^{\infty}$ are dense in $\mathcal{H}_1$.

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['maindense']}
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['maindense2']}