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We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the $L^{p}$ -metric, where $0 < p < 1$ . The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in $H^{p}$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^{p}$ -metric, where $0 < p < 1$ .

On a generalization of Lipschitz's classes.

Krasniqi, Xh.Z. (2008)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On a Monotonic Trigonometric Sum.

Gavin Brown, St. Koumandos (1997)

Monatshefte für Mathematik

On belonging of trigonometric series to Orlicz space.

Tikhonov, S. (2004)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On integrability of functions defined by trigonometric series.

Leindler, Laszlo (2008)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On $L_{1}$ convergence of certain cosine sums.

Braha, N.L., Krasniqi, Xh.Z. (2009)

Bulletin of Mathematical Analysis and Applications [electronic only]

On $L_{1}$ convergence of certain cosine sums with twice quasi semi-convex coefficients.

Braha, N.L. (2010)

APPS. Applied Sciences

On $L^{p}$ integrability and convergence of trigonometric series

Dansheng Yu, Ping Zhou, Songping Zhou (2007)

Studia Mathematica

We first give a necessary and sufficient condition for $x^{- γ} ϕ (x) \in L^{p}$ , 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either $\sum_{k = 1}^{\infty} a_{k} c o s k x$ or $\sum_{k = 1}^{\infty} b_{k} s i n k x$ , under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x) in $L^{p}$ norm.

On some transforms of trigonometric series.

Berisha, F.M., Filipović, M.H. (1997)

Publications de l'Institut Mathématique. Nouvelle Série

On the behavior of $r$ -derivative near the origin of sine series with convex coefficients.

Krasniqi, Xh.Z., Braha, N.L. (2007)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On the integrability and L¹-convergence of double trigonometric series

Ferenc Móricz (1991)

Studia Mathematica

On the integrability and L¹-convergence of sine series

Ferenc Móricz (1989)

Studia Mathematica

On the $L_{1}$ -convergence of Fourier series

S. Fridli (1997)

Studia Mathematica

Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier series...

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

Let a single sine series (*) $\sum_{k = 1}^{\infty} a_{k} s i n k x$ be given with nonnegative coefficients $a_{k}$ . If $a_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k a_{k} \to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $\sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} c_{k l} s i n k x s i n l y$ , even with complex coefficients $c_{k l}$ . We also give a uniform...

On the uniform convergence of weighted trigonometric series

Bogdan Szal (2011)

Banach Center Publications

In the present paper we consider a new class of sequences called GM(β,r), which is the generalization of a class defined by Tikhonov in [15]. We obtain sufficient and necessary conditions for uniform convergence of weighted trigonometric series with (β,r)-general monotone coefficients.

On the utility of the Telyakovskiĭ's class $S$ .

Leindler, L. (2001)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

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