Integrability, mean convergence, and parseval's formula for double trigonometric series

Chang Pao Chen, Chin Cheng Lin

Research output: Contribution to journalArticlepeer-review


Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order (p, 0), (0, p), and (p, p) with the weight (|j|̄ |k|̄)p-1 for some p > 1. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted Lr-integrability and weighted Lr-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], Móricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].

Original languageEnglish
Pages (from-to)191-212
Number of pages22
JournalTaiwanese Journal of Mathematics
Issue number2
StatePublished - Jun 1998


  • Conditions of bounded variation
  • Double trigonometric series
  • Parseval's formula
  • Rectangular partial sums
  • Regular convergence
  • Uniform convergence
  • Weighted L-convergence
  • Weighted L-integrability


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