# On the uniform convergence of double sine series

Studia Mathematica (2009)

- Volume: 193, Issue: 1, page 79-97
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topPéter Kórus, and Ferenc Móricz. "On the uniform convergence of double sine series." Studia Mathematica 193.1 (2009): 79-97. <http://eudml.org/doc/286433>.

@article{PéterKórus2009,

abstract = {Let a single sine series (*) $∑^\{∞\}_\{k=1\} a_\{k\} sin kx$ 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 $ka_\{k\} → 0$ 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 (**) $∑^\{∞\}_\{k=1\} ∑^\{∞\}_\{l=1\} c_\{kl\} sin kx sin ly$, even with complex coefficients $\{c_\{kl\}\}$. We also give a uniform boundedness test for the rectangular partial sums of series (**), and slightly improve the results on single sine series.},

author = {Péter Kórus, Ferenc Móricz},

journal = {Studia Mathematica},

keywords = {double sine series; convergence in Pringsheim's sense; regular convergence; uniform convergence; uniform boundedness; mean value bounded variation double sequence; non-onesided bounded variation double sequence},

language = {eng},

number = {1},

pages = {79-97},

title = {On the uniform convergence of double sine series},

url = {http://eudml.org/doc/286433},

volume = {193},

year = {2009},

}

TY - JOUR

AU - Péter Kórus

AU - Ferenc Móricz

TI - On the uniform convergence of double sine series

JO - Studia Mathematica

PY - 2009

VL - 193

IS - 1

SP - 79

EP - 97

AB - Let a single sine series (*) $∑^{∞}_{k=1} a_{k} sin kx$ 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 $ka_{k} → 0$ 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 (**) $∑^{∞}_{k=1} ∑^{∞}_{l=1} c_{kl} sin kx sin ly$, even with complex coefficients ${c_{kl}}$. We also give a uniform boundedness test for the rectangular partial sums of series (**), and slightly improve the results on single sine series.

LA - eng

KW - double sine series; convergence in Pringsheim's sense; regular convergence; uniform convergence; uniform boundedness; mean value bounded variation double sequence; non-onesided bounded variation double sequence

UR - http://eudml.org/doc/286433

ER -

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.