Approximation by double Walsh polynomials

We study the rate of approximation by rectangular partial sums, Cesà ro 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 L p ( I 2 ) , where 1 ≦ p < ∞ and I 2 = [ 0 , 1 ) × [ 0 , 1 ) , or C W ( I 2 ) , the latter being the collection of uniformly W -continuous functions on I 2 . 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 L p ( I 2 ) -norm and uniform convergence on I 2 as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.