ON ABSOLUTE CONVERGENCE OF MULTIPLE FOURIER SERIES BY S. MINAKSHISUNDARAM AND OTTO SZÂSZ 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/(x) and series of the type^| cn\ ß,ß>0, where the c„ 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 [l ] in the study of summation of multiple Fourier series. A particular result is: if a«,.. .„, are the Fourier coefficients of f(xi, • • • ,x,), and / satisfies a Lipschitz condition of degree a, then 23la»i- • -»«I ß<co for /3>2/c/(«+2a), while the series may be divergent ior ß = 2k/(k+2o). For some previous results concerning the absolute convergence of double Fourier series cf. [3]. 1. Notations. We denote by capital letters vectors in the «-dimensional space, so that X = (xu xt, • ■ ■ , x,), N=(nu «2, • • • , nK); | iV| = £i«2)I/2 is the norm of N; NX =^J[nrx, is the scalar product of N and X. The Xi, • • • , XM are real variables, the »i, • • • , «« are integers. f(xit • • - , xK)=f(X) is a real- valued integrable function of period 2ir in each variable. The formal Fourier series of f(X) is _00 00 (1.1) /(AO ~ Z • • • IX. • • • ,«,e«™+ ■ ■ •+»«*«> = Z cNe°™, where (1.2) c„ = -i_r... C'f(X)e-^dX. (2jr)' J _, J -, Jv(x) is the Bessel function of order /i^O: (z/2)"+2' /,(*) = E (- i)' v\T(p +v+l) we put 2T(m + l)/„(*) " ,. **r(M+i) «/.(*) =-= 2-, (- i)' x» _o 4*v\T(ii + v + 1) Presented to the Society, April 27, 1946; received by the editors April 15, 1946. Í1) Numbers in brackets refer to the bibliography at the end of the paper. 36 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use