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

Ferenc Móricz

Studia Mathematica (1992)

  • Volume: 102, Issue: 3, page 225-237
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Móricz, Ferenc. "On the uniform convergence and L¹-convergence of double Walsh-Fourier series." Studia Mathematica 102.3 (1992): 225-237. <http://eudml.org/doc/215925>.

@article{Móricz1992,
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.},
author = {Móricz, Ferenc},
journal = {Studia Mathematica},
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},
language = {eng},
number = {3},
pages = {225-237},
title = {On the uniform convergence and L¹-convergence of double Walsh-Fourier series},
url = {http://eudml.org/doc/215925},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Móricz, Ferenc
TI - On the uniform convergence and L¹-convergence of double Walsh-Fourier series
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 3
SP - 225
EP - 237
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/215925
ER -

References

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  1. [1] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. Zbl0036.03604
  2. [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). Zbl0502.42006
  3. [3] G. H. Hardy, On double Fourier series, Quart. J. Math. 37 (1906), 53-79. Zbl36.0501.02
  4. [4] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, 1934. 
  5. [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. Zbl53.0226.01
  6. [6] F. Móricz, Approximation by double Walsh polynomials, Internat. J. Math. Math. Sci., to appear. Zbl0817.42014
  7. [7] C. W. Onneweer, On uniform convergence for Walsh-Fourier series, Pacific J. Math. 34 (1970), 117-122. Zbl0205.37602
  8. [8] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279. Zbl0005.24901
  9. [9] R. Salem, Essais sur les séries trigonométriques, Actualités Sci. Indust. 862, Hermann, Paris 1940. Zbl66.0280.01
  10. [10] F. Schipp, W. R. Wade, and P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, Budapest 1990. 
  11. [11] J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 55 (1923), 5-24. Zbl49.0293.03
  12. [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. Zbl0016.10501
  13. [13] A. Zygmund, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959. Zbl0085.05601

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