Measurable functions and spherical summability of multiple Fourier series

Abstract

I. Introduction. This paper is concerned with two problems on the structure of measurable functions which have been completely solved for functions of one variable but only partly solved for several variables. Theorems 1 and 2 below illustrate the usefulness of spherical summability of Fourier series of almost periodic functions and show that much of what was obtained for one variable by complicated and difficult constructions is given in greater generality by using an important result on spherical summability. For the history of the problems considered and some surprising applications of the results, cf. [3], [4], and [1]. II. Statement of the results. The results of Theorems 1 and 2 can be proved for functions with values in an arbitrary Banach space. To avoid confusion the results will be stated for real-valued functions; and the proof of Theorem 1 will be given only for two variables, i.e., for the square J2, (J2= [-r, 1T] X [-r, 1r] Jk iS the product of k copies of the interval [-oir, or]). THEOREM 1. Let f(xl, . . ., Xk) be any function on Jk which is measurable and finite valued almost everywhere. There exists a continuous additive interval function F(I) such that almost everywhere F'(x1, . . ., xj) =f(x1, . . ., Xk) and a trigonometric series : a.,,7, exp [(nix, + * * + nkxk)] which is summable with sum f(xl,..., Xk) almost everywhere by means of any summation function of type (k, k).