Geodesic
On the uniform convergence and L¹-convergence of double Walsh-Fourier series
Studia Mathematica, Tome 102 (1992) no. 3, pp. 225-237 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, 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.
DOI : 10.4064/sm-102-3-225-237
Keywords: Walsh-Paley system, W-continuity, moduli of continuity and smoothness, bounded variation in the sense of Hardy and Krause, generalized bounded variation, complementary functions in the sense of W. H. Young, rectangular partial sum, Dirichlet kernel, convergence in $L^p$-norm, uniform convergence Salem's test, Dini-Lipschitz test, Dirichlet-Jordan test

Ferenc Móricz 1

1 
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Ferenc Móricz. On the uniform convergence and L¹-convergence of double Walsh-Fourier series. Studia Mathematica, Tome 102 (1992) no. 3, pp. 225-237. doi: 10.4064/sm-102-3-225-237

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