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On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system. (English) Zbl 1236.42024

Let \(f\in L^1([0,1)^2)\) and denote by \(S_{k,\ell}f\) the rectangular partial sums of the expansion of \(f\) with respect to the two-dimensional Walsh-Paley system. The Marcinkiewicz-like means of \(f\) are defined by \[ t^\alpha_n f:={1\over n} \sum^{n-1}_{k=0} S_{\alpha(|n|,k)}f,\quad n= 1,2,\dots, \] where \(\alpha: \mathbb{N}^2\to \mathbb{N}^2\) and \(|n|\) is the lower integer part of the binary logarithm of \(n\). In his seminal paper [Ann. Soc. Polon. Math. 16, 85–96 (1937); cf. J. Marcinkiewicz and A. Zygmund, Fundam. Math. 28, 309–335 (1936; Zbl 0016.20502; JFM 63.0202.01)], J. Marcinkiewicz investigated the case when \(\alpha(|n|,k):= (k,k)\). The present author gives a kind of necessary and sufficient condition in order that the limit \(t^\alpha_n f\to f\) exist almost everywhere for every \(f\in L^1([0,1)^2)\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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