# Uniform convergence of double trigonometric series

Studia Mathematica (1996)

• Volume: 118, Issue: 3, page 245-259
• ISSN: 0039-3223

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## Abstract

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It is shown that under certain conditions on ${c}_{jk}$, the rectangular partial sums ${s}_{mn}\left(x,y\right)$ converge uniformly on ${T}^{2}$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is ${\sum }_{|k|=n}^{\infty }|\Delta {c}_{k}|=o\left(1/n\right)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $n{c}_{n}=o\left(1\right)$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].

## How to cite

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Chen, Chang-Pao, and Chen, Gwo-Bin. "Uniform convergence of double trigonometric series." Studia Mathematica 118.3 (1996): 245-259. <http://eudml.org/doc/216276>.

@article{Chen1996,
abstract = {It is shown that under certain conditions on $\{c_\{jk\}\}$, the rectangular partial sums $s_\{mn\}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is $∑_\{|k|= n\}^∞ |Δc_k| = o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $nc_\{n\} = o(1)$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].},
author = {Chen, Chang-Pao, Chen, Gwo-Bin},
journal = {Studia Mathematica},
keywords = {uniform convergence; double trigonometric series; rectangular partial sums},
language = {eng},
number = {3},
pages = {245-259},
title = {Uniform convergence of double trigonometric series},
url = {http://eudml.org/doc/216276},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Chen, Chang-Pao
AU - Chen, Gwo-Bin
TI - Uniform convergence of double trigonometric series
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 3
SP - 245
EP - 259
AB - It is shown that under certain conditions on ${c_{jk}}$, the rectangular partial sums $s_{mn}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is $∑_{|k|= n}^∞ |Δc_k| = o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $nc_{n} = o(1)$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].
LA - eng
KW - uniform convergence; double trigonometric series; rectangular partial sums
UR - http://eudml.org/doc/216276
ER -

## References

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1. [CJ] T. W. Chaundy and A. E. Jolliffe, The uniform convergence of a certain class of trigonometric series, Proc. London Math. Soc. (2) 15 (1916), 214-216.
2. [C1] C.-P. Chen, Weighted integrability and ${L}^{1}$-convergence of multiple trigonometric series, Studia Math. 108 (1994), 177-190. Zbl0821.42007
3. [C2] C.-P. Chen, Integrability of multiple trigonometric series and Parseval's formula, J. Math. Anal. Appl. 186 (1994), 182-199. Zbl0807.42007
4. [CL] C.-P. Chen and C.-C. Lin, Integrability, mean convergence, and Parseval's formula for double trigonometric series, preprint. Zbl0907.42009
5. [D] M. I. Dyachenko, The rate of u-convergence of multiple Fourier series, Acta Math. Hungar. 68 (1995), 55-70. Zbl0828.42007
6. [J] A. E. Jolliffe, On certain trigonometric series which have a necessary and sufficient condition for uniform convergence, Math. Proc. Cambridge Philos. Soc. 19 (1921), 191-195.
7. [K] J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53. Zbl56.0907.01
8. [M1] F. Móricz, Convergence and integrability of double trigonometric series with coefficients of bounded variation, Proc. Amer. Math. Soc. 102 (1988), 633-640. Zbl0666.42004
9. [M2] F. Móricz, On the integrability and ${L}^{1}$-convergence of double trigonometric series, Studia Math. 98 (1991), 203-225. Zbl0724.42015
10. [M3] F. Móricz, On the integrability of double cosine and sine series I, J. Math. Anal. Appl. 154 (1991), 452-465. Zbl0724.42013
11. [N] J. R. Nurcombe, On the uniform convergence of sine series with quasimonotone coefficients, ibid. 166 (1992), 577-581. Zbl0756.42006
12. [S] O. Szász, Quasi-monotone series, Amer. J. Math. 70 (1948), 203-206. Zbl0035.03901
13. [XZ] T. F. Xie and S. P. Zhou, The uniform convergence of certain trigonometric series, J. Math. Anal. Appl. 181 (1994), 171-180. Zbl0791.42004
14. [Z] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1968. Zbl0157.38204

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