# On the uniform convergence of double sine series

Studia Mathematica (2009)

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

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## Abstract

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Let a single sine series (*) ${\sum }_{k=1}^{\infty }{a}_{k}sinkx$ 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 (**) ${\sum }_{k=1}^{\infty }{\sum }_{l=1}^{\infty }{c}_{kl}sinkxsinly$, 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.

## How to cite

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Péter 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 -

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