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1992 | 102 | 3 | 225-237
Tytuł artykułu

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

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Abstrakty
EN
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.
Twórcy
  • Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary
Bibliografia
  • [1] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414.
  • [2] 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).
  • [3] G. H. Hardy, On double Fourier series, Quart. J. Math. 37 (1906), 53-79.
  • [4] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, 1934.
  • [5] 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.
  • [6] F. Móricz, Approximation by double Walsh polynomials, Internat. J. Math. Math. Sci., to appear.
  • [7] C. W. Onneweer, On uniform convergence for Walsh-Fourier series, Pacific J. Math. 34 (1970), 117-122.
  • [8] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279.
  • [9] R. Salem, Essais sur les séries trigonométriques, Actualités Sci. Indust. 862, Hermann, Paris 1940.
  • [10] F. Schipp, W. R. Wade, and P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, Budapest 1990.
  • [11] J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 55 (1923), 5-24.
  • [12] 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.
  • [13] A. Zygmund, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959.
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bwmeta1.element.bwnjournal-article-smv102i3p225bwm
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