Almost periodicity and convergent trigonometric series

Abstract

1. Introduction. Besicovitch [1 ] has shown that there are infinitely many trigonometric series convergent to any function f(x) bounded and continuous for - o x < + oo and of bounded variation in every finite interval, and therefore that the sum-function of an everywhere convergent trigonometric series is not necessarily almost periodic. His proof makes clear that almost periodicity of the sum-function can only be ensured by placing a suitable restriction on the exponents of the series. The conditions to be satisfied by the sum-function itself depend on the type of almost periodicity to be established. By use of the symmetrical (SCP) and unsymmetrical (CP) Cesaro-Perron integrals and the introduction of new distance-functions, H. Burkill [4] has enlarged the class of almost periodic functions sufficiently to enable him to obtain the following result, whose most remarkable feature is that no assumption is made concerning the integrability of the sum-f unction. It is a generalization of a similar theorem for purely periodic functions, due to J. C. Burkill [5]. H. BURKILL'S THEOREM. If (X.) be a sequence of real numbers such that