L(1)-CONVERGENCE OF DOUBLE COSINE AND WALSH-FOURIER SERIES

We prove the convergence in L(1)([-pi, pi)(2))-, of the double Fourier series of an integrable function f(x,y) which is periodic and even with respect to x and y, with coefficients a(jk) satisfying certain conditions of Hardy-Karamata kind, and such that a(jk) log j log k --> 0 as j, k --> infinity. These sufficient conditions become quite natural In particular cases. Then we extend these results to the convergence of double Walsh-Fourier series in L(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.