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131 261 263 Gát, G. , On ( C , 1) summability of integrable functions with respect to the Walsh-Kaczmarz system, Studia Math

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Abstract  

Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H p to the space L p for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ * α,k is bounded from the martingale Hardy space H 1/(1+α) to the space weak-L 1/(1+α).

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The main aim of this paper is to prove that the maximal operator σ 0 k*:= supnσ n,n k∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H 1/2 to the space weak-L 1/2.

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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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, U. , A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh–Kaczmarz system , Acta Math. Hungar. , 125 ( 2009 ), no. 1–2 , 65 – 83 . [9

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series with respect to the Walsh-Kaczmarz system, Studia Sci. Math. Hungar. , 46 (2009), no. 3, 399–421. Nagy K. On the Marcinkiewicz-Fejér means of double Fourier series with

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] Gát , G. , Goginava , U. and Nagy , K. , On the Marcinkiewicz-Fejér means of double Fourier series with respect to Walsh-Kaczmarz system . Studia Sci. Math. Hungar. , 46 : 399 – 421 , 2009 . [11

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