Approximation in the mean on rational curves

TL;DR

This paper explores the relationship between polynomial density in Lebesgue spaces and bounded point evaluations on rational curves, extending classical complex plane results to algebraic curves.

Contribution

It establishes new connections between polynomial approximation and analytic evaluations on rational algebraic curves, generalizing known complex plane theorems.

Findings

Polynomial density linked to bounded point evaluations on rational curves

Analogues of Thomson and Brennan's results extended to algebraic curves

Provides a framework for approximation theory on rational algebraic varieties

Abstract

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves.