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Abstract  

By constructing an example, the present note will show that the condition
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raised by Tikhonov [2] to study L convergence case cannot keep the classical theorem working in L 1 convergence case.
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Bray, W. O. and Stanojevic, Č. V. , On weighted integrability of trigonometric series and L 1 -convergence of Fourier series, Proc. Amer. Math. Soc. 96 (1986), 53–61. MR 87e :42007

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The present note will present a different and direct way to generalize the convexity while keep the classical results for L 1-convergence still alive.

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References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar

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Abstract  

Chaundry and Jolliffe [1] proved that if a k is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series Σk=1 a k sin kx is limk→∞ ka k = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L 1-convergence of Fourier series.

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References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar

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Abstract  

Our aim is to find the source why the logarithm sequences play the crucial role in the L 1-convergence of sine series. We define three new classes of sequences; one of them has the character of the logarithm sequences, the other two are the extensions of the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences. In terms of these classes, extended analogues of Zhou’s theorems are proved.

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References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar

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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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Fourier Analysis on Local Fields Mathematical Notes Princeton Universitiy Press Princeton . [16] Tanović-Miller , N. 2003 Integrability and L 1

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