Generalized absolute continuity of a function of Wiener's class

Abstract

In the present paper we give a criterion for a function of Wiener's class to belong to the class of generalized absolute continuity, in terms of Fourier-Young coefficients {Ck}. More precisely, we prove the following theorem.THEOREM. Let Λ = (λn,k) be a normal almost periodic matrix of real numbers such that λn,k ≥ λn,k+1 for all n and k. Then for any function f of Wiener's class Vν (1 < ν < 2) to be of class of generalized absolute continuity Ap (1 < p < ∞) it is necessary and sufficient that {|Ck|2} is summable Λ to zero.