On absolute convergence of multiple Fourier series

Abstract

Introduction. The results of this paper are extensions of corresponding results for simple Fourier series, given by one of the authors (cf. [5 ]) (1). The main problem was to study the relationship between the mean modulus of a function f(x) and series of the type E Icn I , /3 >0, where the cn are the Fourier coefficients of f(x). We obtain here analoguous results, employing spherical means of a function of several variables. These means were first used by Bochner [1 ] in the study of summation of multiple Fourier series. A particular result is: if a,,, . . .nn are the Fourier coefficients of f(xi, x,x) and f satisfies a Lipschitz condition of degree a, then |iI a., ... .jI 2KI/(K +2a), while the series may be divergent for /3= 2 K/(K+2a). For some previous results concerning the absolute convergence of double Fourier series cf. [3]. 1. Notations. We denote by capital letters vectors in the K-dimensional space, so that X = (x1, X2, * xi), N = (n1, n2, * ni); I NJ = ( ',n2)1I2 is the norm of N; NX =En'x is the scalar product of N and X. The xi, *, are real variables, the ni, * * *, n are integers. f(x1, * * * , XK) =f(X) is a realvalued integrable function of period 27r in each variable. The formal Fourier series of f(X) is