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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weighted integrability
of double trigonometric series


Authors: Chang-Pao Chen and Xin-Rong Huang
Journal: Proc. Amer. Math. Soc. 127 (1999), 1463-1471
MSC (1991): Primary 42B99, 42A16
Published electronically: January 29, 1999
MathSciNet review: 1476123
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Abstract: We study the double trigonometric series whose coefficients $c_{jk}$ are such that $\sum _{j=-\infty}^\infty\sum _{k=-\infty}^\infty |c_{jk}|<\infty.$ Then its rectangular partial sums converge uniformly to some $f\in C(T^2)$. We give sufficient conditions for the Lebesgue integrability of $\{f(x,y)-f(x,0)-f(0,y)+f(0,0)\}\phi(x,y)$, where $\phi(x,y)=1/xy, 1/x$, or $1/y$. For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.


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Additional Information

Chang-Pao Chen
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Email: cpchen@math.nthu.edu.tw

Xin-Rong Huang
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04661-4
PII: S 0002-9939(99)04661-4
Received by editor(s): September 1, 1997
Published electronically: January 29, 1999
Additional Notes: This research was supported by National Science Council, Taipei, R.O.C., under Grant #NSC 86-2115-M-007-012.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society