Approximation in the Mean by Polynomials

Abstract

For a positive measure AL with compact support in the complex plane C and for 1 < t < co, let pt(i) denote the closure in Lt(/i) of the polynomials in one complex variable. The purpose of this paper is to describe pt(A). The main results are the following. First, there exists a Borel partition {Ail' = of the support of Al such that for i ? 1 the space Pt(t LAAi) contains no nontrivial characteristic functions and