Pointwise Convergence of Fourier Series

Abstract

In this paper, we present a new proof of a theorem of Carleson and Hunt: The Fourier series of an LP function on [0, 2J] converges almost everywhere (p > 1). (See [1], [51.) Our proof is very much in the spirit of the classical theorem of Kolmogoroff-Seliverstoff-Plessner [8]. Unlike Carleson's proof, which makes a careful analysis of the structure of an L2 function f, our arguments essentially ignore f, and concentrate instead on building up a basic partial sum operator from simpler pieces. Our methods are (almost) entirely L2. Sections 1-7 of this paper contain a proof of pointwise convergence for L2 functions; Section 8 contains the modifications necessary to handle LP, and includes various further comments.