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References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar
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 Σ k=1 ∞ a k sin kx is lim k→∞ 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.
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.
In the present paper we give a brief review of L 1 -convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.
Abstract
Abstract
We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from “one sided” to “two sided”, while some important classical results keep true.
References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar
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.
References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar
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 f∈L 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.
