Abstract
For a non-negative integer n let us denote the dyadic variation of a natural number n by
where n := ∑ ∞i=0 n i 2i, n i ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition sup A V (nA) < ∞, the subsequence of quadratic partial sums \(S{_n^{\square}}_A\left( f \right)\) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {n A : A ≥ 1} with the condition sup A V(nA) < ∞ and a function f ∈ φ(L)(I2) for which \(\text{sup} _A|S{_n^{\square}}_A\left( {{x^1},{x^2};f} \right)| = \infty \) for almost all (x1, x2) ∈ I2.
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References
G. Gát, On the divergence of the (C, 1) means of double Walsh–Fourier series, Proc. Amer. Math. Soc., 128 (2000), 1711–1720.
G. Gát, On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system, J. Approx. Theory, 164 (2012), 145–161.
R. Gosselin, On the divergence of Fourier series, Proc. Amer. Math. Soc., 9 (1958), 278–282.
S. V. Konyagin, On a subsequence of Fourier–Walsh partial sums, Mat. Zametki, 54 (1993), 69–75 (in Russian); translated in Math. Notes, 54 (1993), 1026–1030.
M. A. Krasnosel’skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces (Moscow, 1958) (in Russian); English translation: P. Noordhoff Ltd. (Groningen, 1961).
S. F. Lukomskiĭ, On the divergence almost everywhere of Fourier-Walsh quadratic partial sums of integrable functions, Mat. Zametki, 56 (1994), 57–62 (in Russian); translated in Math. Notes, 56 (1994), 690–693.
S. F. Lukomskiĭ, A criterion for the convergence almost everywhere of Fourier-Walsh quadratic partial sums of integrable functions, Mat. Sb., 186 (1995), 133–146 (in Russian); translated in Sb. Math., 186 (1995), 1057–1070.
S. F. Lukomskii, Convergence of multiple Walsh series in measure and in L, East J. Approx., 3 (1997), 317–332.
F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger (Bristol, New York, 1990).
V. Totik, V. On the divergence of Fourier series, Publ. Math. Debrecen, 29 (1982), 251–264.
F. Weisz, Summability of Multi-dimensional Fourier Series and Hardy Space, Kluwer Academic Publishers (Dordrecht, 2002).
A. Zygmund, Trigonometric Series, vol. 1, Cambridge University Press (1959).
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The first author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K111651 and by project EFOP-3.6.2-16-2017-00015 supported by the European Union, cofinanced by the European Social Fund.
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Gát, G., Goginava, U. Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series. Anal Math 44, 73–88 (2018). https://doi.org/10.1007/s10476-018-0107-2
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DOI: https://doi.org/10.1007/s10476-018-0107-2