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International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 2, Pages 209-220

Approximation by double Walsh polynomials

University of Szeged, Bolyai Institute, Aradi vértanúk tere 1, Szeged 6720, Hungary

Received 20 June 1991

Copyright © 1992 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 1p< 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.