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Gát, G. , Cesáro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory , 124 (2003), no. 1, 25–43. Gát G. Cesáro means of integrable

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Gát, G. , Convergence of Marcinkiewicz means of integrable functions with respect to two-dimensional Vilenkin system, Georg. Math. J. 10(3) (2004), 467–478. MR 2005h :42059 Gát G

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Vilenkin systems , J. Approx. Theory , 124 ( 2003 ), no. 1 , 25 – 43 . [3] Goginava , U. , The maximal operator of the (C, α) means of the Walsh–Fourier series

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In this paper we establish approximation properties of Cesàro (C, −α) means with α ∈ (0, 1) of Vilenkin—Fourier series. This result allows one to obtain a condition which is sufficient for the convergence of the means σ n α(f, x) to f(x) in the L p-metric.

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Abstract  

Characterizations of the two-dimensional H 1, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

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Abstract  

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function fL 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

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