Linear functions on almost periodic functions

Abstract

Our aim is, first, to present two realizations of the space 1* of all bounded complex linear functionals on 21, and, second, to use these realizations for the study of positive definite functions which are not necessarily continuous. In this fashion, we obtain a generalization of Bochner's representation theorem for continuous positive definite functions as well as various facts concerning positive definite functions and their structure. 0.2 Throughout the present paper, the symbol R denotes the real numbers, considered either as an additive group or as a field; K the field of complex numbers; T the multiplicative group of complex numbers of absolute value 1; Tm the complete Cartesian product of m groups each identical with T, m being any cardinal number greater than 1. The characteristic function of a subset B of a set X is denoted by XB. If G is any locally compact Abelian group, we denote the group of all continuous characters of G by the symbol G]. G] is given the usual compact-open topology. If X is any topological space, we denote the set of all complex-valued continuous functions on X which are bounded in absolute value by the symbol C(X). The space of all trigonometric polynomials 0.1.1 is denoted by 93; the space of all almost periodic continuous functions on R by 2f. For a normed complex linear space V, we denote the space of all bounded complex linear functionals on V by the symbol V*.