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

Studia Mathematica (1992)

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

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topMóricz, 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

top- [1] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. Zbl0036.03604
- [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] G. H. Hardy, On double Fourier series, Quart. J. Math. 37 (1906), 53-79. Zbl36.0501.02
- [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. Zbl53.0226.01
- [6] F. Móricz, Approximation by double Walsh polynomials, Internat. J. Math. Math. Sci., to appear. Zbl0817.42014
- [7] C. W. Onneweer, On uniform convergence for Walsh-Fourier series, Pacific J. Math. 34 (1970), 117-122. Zbl0205.37602
- [8] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279. Zbl0005.24901
- [9] R. Salem, Essais sur les séries trigonométriques, Actualités Sci. Indust. 862, Hermann, Paris 1940. Zbl66.0280.01
- [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. Zbl49.0293.03
- [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] A. Zygmund, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959. Zbl0085.05601

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