Power series with almost periodic coefficients

Abstract

and that the corresponding limit at x = e(O), where 0 is not congruent (mod 1) to an integral multiple of a, is zero. P. Sziisz has pointed out that for the proof of (1) and (2) Davenport's conditions can be reduced to the following: (a) limN_. EInIN ane(nt) exist for each real t, and (b) I E jnI,N a.e(nt) I <K < co uniformly in N and t. That is, the convergence of Ean and condition (6) in Davenport's paper can be replaced by (a). The purpose of this note is to generalize the above results in a couple of directions. It is to be noticed that the power series coefficients in (2), A,=g(va+y), constitute in some general sense an almost periodic sequence with the Fourier expansion