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New uniqueness theorems for trigonometric series

Authors: J. Marshall Ash and Sh. T. Tetunashvili
Journal: Proc. Amer. Math. Soc. 128 (2000), 2627-2636
MSC (1991): Primary 42A63, 42B99; Secondary 42A20, 42A24
Published electronically: February 25, 2000
MathSciNet review: 1657746
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Abstract | References | Similar Articles | Additional Information


A uniqueness theorem is proved for trigonometric series and another one is proved for multiple trigonometric series. A corollary of the second theorem asserts that there are two subsets of the $d$-dimensional torus, the first having a countable number of points and the second having $2^d$ points such that whenever a multiple trigonometric series ``converges'' to zero at each point of the former set and also converges absolutely at each point of the latter set, then that series must have every coefficient equal to zero. This result remains true if ``converges'' is interpreted as any of the usual modes of convergence, for example as ``square converges'' or as ``spherically converges.''

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

  • [AW] J. Marshall Ash and Gang Wang, A survey of uniqueness questions in multiple trigonometric series, Harmonic analysis and nonlinear differential equations (Riverside, CA, 1995), Contemp. Math., vol. 208, Amer. Math. Soc., Providence, RI, 1997, pp. 35–71. MR 1467001, 10.1090/conm/208/02734
  • [K] V. Ya. Kozlov, On the connection between absolute convergence and the uniqueness of expansion of a function into trigonometric series, Doklady Akademii Nauk USSR, 15(1937), 417-420.

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Additional Information

J. Marshall Ash
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504

Sh. T. Tetunashvili
Affiliation: Department of Mathematics, Georgian Technical University, Kortava st. 77, Tbilisi, Georgia 380075

Keywords: Trigonometric series, multiple trigonometric series, uniqueness
Received by editor(s): August 1, 1998
Received by editor(s) in revised form: October 16, 1998
Published electronically: February 25, 2000
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society