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

• Autorzy: Ferenc Móricz
• Języki publikacji: EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 41A50: Best approximation, Chebyshev systems
• 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1992
• Tom: 102
• Numer: 3
• Strony: 225-237

Daty

wydano

1992

otrzymano

1991-07-31

Twórcy

autor

Ferenc Móricz

• 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.