Properties of the Coefficients of Orthonormal Sequences

Abstract

In this paper we consider complete orthonormal sequences defined on the interval [0, 1] and satisfying an inequality of the type(1)for allnand some sequence {Fn}. Such sequences were first considered by Zygmund and Marcinkiewicz (8). They extended the well-known results of Hausdorff-Young and Paley, originally proved for the casev= ∞,Fn=Mfor alln(12). We will consider cases of equality in t he Hausdorff-Young theorems and certain limiting cases of the Paley theorems. Application of these results and the results in (8) will be made to functions harmonic in the unit α-sphere.