Annals of Functional Analysis

On an approximation of 2-dimensional Walsh–Fourier series in martingale Hardy spaces

L. E. Persson, G. Tephnadze, and P. Wall

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In this paper, we investigate convergence and divergence of partial sums with respect to the 2-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh–Fourier series. We also show that these conditions are in a sense necessary and sufficient.

Article information

Ann. Funct. Anal., Volume 9, Number 1 (2018), 137-150.

Received: 29 November 2016
Accepted: 11 March 2017
First available in Project Euclid: 5 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory

Fourier series Walsh system strong summability martingale Hardy space 2-dimensional modulus of continuity


Persson, L. E.; Tephnadze, G.; Wall, P. On an approximation of $2$ -dimensional Walsh–Fourier series in martingale Hardy spaces. Ann. Funct. Anal. 9 (2018), no. 1, 137--150. doi:10.1215/20088752-2017-0032.

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  • [1] G. N. Agaev, N. Y. Vilenkin, G. M. Dzhafarly, and A. I. Rubinstein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups, Èlm, Baku, 1981.
  • [2] I. Blahota, On a norm inequality with respect to Vilenkin-like systems, Acta Math. Hungar. 89 (2000), no. 1–2, 15–27.
  • [3] S. Fridli, P. Manchanda, and A. H. Siddiqi, Approximation by Walsh–Nörlund means, Acta Sci. Math. (Szeged) 74 (2008), no. 3–4, 593–608.
  • [4] G. Gát, Investigations of certain operators with respect to the Vilenkin system, Acta Math. Hungar. 61 (1993), no. 1–2, 131–149.
  • [5] U. Goginava and L. D. Gogoladze, “Strong convergence of cubic partial sums of two-dimensional Walsh–Fourier series” in Constructive Theory of Functions (Sozopol, 2010), Prof. Marin Drinov Acad. Publ., Sofia, 2012, 108–117.
  • [6] B. Golubov, A. Efimov, and V. Skvortsov, Walsh Series and Transforms: Theory and Applications, Math. Appl. (Soviet Series) 64, Kluwer Acad., Dordrecht, 1991.
  • [7] Y. Jiao, D. Zhou, Z. Hao, and W. Chen, Martingale Hardy spaces with variable exponents, Banach J. Math. Anal. 10 (2016), no. 4, 750–770.
  • [8] N. Memić, I. Simon, and G. Tephnadze, Strong convergence of two-dimensional Vilenkin–Fourier series, Math. Nachr. 289 (2016), no. 4, 485–500.
  • [9] K. Nagy and G. Tephnadze, The Walsh–Kaczmarz–Mancinkiewicz means and Hardy spaces, Acta Math. Hungar. 149 (2016), no. 2, 346–374.
  • [10] F. Schipp, W. R. Wade, P. Simon, and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  • [11] P. Simon, Strong convergence of certain means with respect to the Walsh–Fourier series, Acta Math. Hungar. 49 (1987), no. 3–4, 425–431.
  • [12] P. Simon, Strong convergence theorem for Vilenkin–Fourier series, J. Math. Anal. Appl. 245 (2000), no. 1, 52–68.
  • [13] G. Tephnadze, A note on the Fourier coefficients and partial sums of Vilenkin–Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 28 (2012), no. 2, 167–176.
  • [14] G. Tephnadze, Strong convergence of two-dimensional Walsh–Fourier series (in Russian), Ukraïn. Mat. Zh. 65 (2013), no. 6, 822–834; English translation in Ukranian Math. J. 65 (2013), 914–927.
  • [15] G. Tephnadze, On the partial sums of Vilenkin–Fourier series (in Russian), Izv. Nats. Akad. Nauk. Armenii Mat. 49 (2014), no. 1, 60–72; English translation in J. Contemp. Math. Anal. 49 (2014), 23–32.
  • [16] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [17] F. Weisz, Cesàro summability of one- and two-dimensional Walsh–Fourier series, Anal. Math. 22 (1996), no. 3, 229–242.
  • [18] F. Weisz, Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric Fourier series, Studia Math. 117 (1996), no. 2, 173–194.
  • [19] F. Weisz, Summability of Multi-Dimensional Fourier Series and Hardy Spaces, Math. Appl. 541, Kluwer Acad., Dordrecht, 2002.
  • [20] F. Weisz, Triangular summability and Lebesgue points of 2-dimensional Fourier transforms, Banach J. Math. Anal. 11 (2017), no. 1, 223–238.