Absolute convergence of multiple Fourier series of functions of $$\phi \left( \varLambda ^1,\ldots ,\varLambda ^N\right) $$ ϕ Λ 1 , … , Λ N -bounded variation

Abstract

In this paper, we provide a sufficient condition for the absolute convergence of multiple Fourier series of a function of φ Λ 1 , . . . , Λ N -bounded variation. This condition generalizes one for a single Fourier series proved by Schramm and Waterman. We apply the classical technique of summation by parts and some key inequalities for convex functions (including the Jensen's inequality) and Δ 2 -functions to prove our main result.