L 1-convergence of double cosine- and Walsh-Fourier series

Abstract

We prove the convergence inL 1([−gp, π)2)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa jk satisfying certain conditions of Hardy-Karamata kind, and such thata jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL 1 (0, 1)2)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesaro summable.