Abstract

We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach space X. In particular, X may be Lp(I2), where 1≦p<∞ and I2=[0,1)×[0,1), or CW(I2), the latter being the collection of uniformly W-continuous functions on I2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence in Lp(I2)-norm and uniform convergence on I2 as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.