Sets of uniqueness for spherically convergent multiple trigonometric series
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- by J. Marshall Ash and Gang Wang PDF
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Abstract:
A subset $E$ of the $d$-dimensional torus $\mathbb {T}^{d}$ is called a set of uniqueness, or $U$-set, if every multiple trigonometric series spherically converging to $0$ outside $E$ vanishes identically. We show that all countable sets are $U$-sets and also that $H^{J}$ sets are $U$-sets for every $J$. In particular, $C\times \mathbb {T}^{d-1}$, where $C$ is the Cantor set, is an $H^{1}$ set and hence a $U$-set. We will say that $E$ is a $U_{A}$-set if every multiple trigonometric series spherically Abel summable to $0$ outside $E$ and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for $U_{A}$ sets. In addition, every $U_{A}$-set has measure $0$, and a countable union of closed $U_{A}$-sets is a $U_{A}$-set.References
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Additional Information
- J. Marshall Ash
- Affiliation: Mathematics Department, DePaul University, Chicago, Illinois 60614
- MR Author ID: 27660
- Email: mash@math.depaul.edu
- Gang Wang
- Affiliation: Mathematics Department, DePaul University, Chicago, Illinois 60614
- Email: gwang@math.depaul.edu
- Received by editor(s): July 15, 1999
- Received by editor(s) in revised form: April 24, 2000
- Published electronically: July 25, 2002
- Additional Notes: This paper is in final form and no version of it will be submitted for publication elsewhere
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4769-4788
- MSC (2000): Primary 05C38, 15A15; Secondary 05A15, 15A18
- DOI: https://doi.org/10.1090/S0002-9947-02-03086-6
- MathSciNet review: 1926836