ON THE UNIFORM-CONVERGENCE AND L1-CONVERGENCE OF DOUBLE WALSH-FOURIER SERIES

In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh Fourier series in a more general setting. We study the convergence of rectangular partial sums in L(p)-norm for some 1 less-than-or-equal-to p less-than-or-equal-to infinity over the unit square [0, 1) x [0, 1). In case p = infinity by L(p) we mean C(W), the collection of uniformly W-continuous functions f(x, y), endowed with the supremum norm. As special cases, we obtain the extensions of the Dini Lipschitz test and the Dirichlet-Jordan test for double Walsh-Fourier series.