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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
DOI: https://doi.org/10.1090/S0002-9939-99-04661-4
Published electronically: January 29, 1999
MathSciNet review: 1476123
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [Ba] N. K. Bary, A Treatise on Trigonometric Series, Pergamon, Oxford, 1964, p. 656.
  • [B1] R. P. Boas, Integrability of trigonometric series, Duke Math. J. 18 (1951), 787-793. MR 13:549b
  • [B2] -, Integrability of trigonometric series III, Quart. J. Math. Oxford Ser. (2) 3 (1952), 217-221. MR 14:867b
  • [B3] -, Integrability Theorems for Trigonometric Transforms, Springer-Verlag, Berlin/ New York, 1967. MR 36:3043
  • [BW] G. Brown and K. Y. Wang, On a conjecture of F. Móricz, Preprint. CMP 97:13
  • [C1] C.-P. Chen, Weighted integrability and $L^{1}$-convergence of multiple trigonometric series, Studia Math. 108 (1994), 177-190. MR 95b:42010
  • [C2] -, Integrability of multiple trigonometric series and Parseval's formula, J. Math. Anal. Appl. 186 (1994), 182-199. MR 95h:42011
  • [Ma] M. M. H. Marzuq, Integrability theorem of multiple trigonometric series, J. Math. Anal. Appl. 157 (1991), 337-345. MR 92f:42008
  • [M1] F. Móricz, Integrability of double lacunary sine series, Proc. Amer. Math. Soc. 110 (1990), 355- 364. MR 90m:42020
  • [M2] -, On the integrability of double cosine and sine series I, J. Math. Anal. Appl. 154 (1991), 452-465. MR 92b:42016
  • [M3] -, Integrability of double cosine-sine series in the sense of improper Riemann integral, J. Math. Anal. Appl. 165 (1992), 419-437. MR 93g:42007
  • [M4] -, Integrability theorems for trigonometric series, Acta Sci. Math. (Szeged) 57 (1993), 443-452.MR 95d:42006
  • [P] K. F. Papp, Lebesgue integrability theorems for double trigonometric series, Acta Sci. Math. (Szeged) 61 (1995), 331-344. MR 97m:42018
  • [Z] A. Zygmund, Trigonometric series (2nd ed.), Cambridge Univ. Press, Cambridge, 1968. MR 38:4882

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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: https://doi.org/10.1090/S0002-9939-99-04661-4
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

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