On the uniform convergence and L¹-convergence of double Walsh-Fourier series

Ferenc Móricz

ArticleOriginal scientific text

Title

On the uniform convergence and L¹-convergence of double Walsh-Fourier series

Authors ¹

Affiliations

Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary

Abstract

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.

Keywords

ENG

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

N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414.
R. D. Getsadze, Convergence and divergence of multiple orthonormal Fourier series in C and L metrics, Soobshch. Akad. Nauk Gruzin. SSR 106 (1982), 489-491 (in Russian).
G. H. Hardy, On double Fourier series, Quart. J. Math. 37 (1906), 53-79.
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, 1934.
E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. 1, third edition, Cambridge Univ. Press, 1927; Dover, New York 1957.
F. Móricz, Approximation by double Walsh polynomials, Internat. J. Math. Math. Sci., to appear.
C. W. Onneweer, On uniform convergence for Walsh-Fourier series, Pacific J. Math. 34 (1970), 117-122.
R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279.
R. Salem, Essais sur les séries trigonométriques, Actualités Sci. Indust. 862, Hermann, Paris 1940.
F. Schipp, W. R. Wade, and P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, Budapest 1990.
J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 55 (1923), 5-24.
L. C. Young, Sur une généralisation de la notion de variation de puissance p-ième bornée au sens de M. Wiener, et sur la convergence des séries de Fourier, C. R. Acad. Sci. Paris 204 (1937), 470-472.
A. Zygmund, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959.

Title

On the uniform convergence and L¹-convergence of double Walsh-Fourier series

Affiliations

Abstract

Keywords

Bibliography