New uniqueness theorems for trigonometric series
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- by J. Marshall Ash and Sh. T. Tetunashvili PDF
- Proc. Amer. Math. Soc. 128 (2000), 2627-2636 Request permission
Abstract:
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
- 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, DOI 10.1090/conm/208/02734
- 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.
Additional Information
- J. Marshall Ash
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504
- MR Author ID: 27660
- Email: mash@math.depaul.edu
- Sh. T. Tetunashvili
- Affiliation: Department of Mathematics, Georgian Technical University, Kortava st. 77, Tbilisi, Georgia 380075
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2627-2636
- MSC (1991): Primary 42A63, 42B99; Secondary 42A20, 42A24
- DOI: https://doi.org/10.1090/S0002-9939-00-05272-2
- MathSciNet review: 1657746