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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sets of uniqueness for spherically convergent multiple trigonometric series


Authors: J. Marshall Ash and Gang Wang
Journal: Trans. Amer. Math. Soc. 354 (2002), 4769-4788
MSC (2000): Primary 05C38, 15A15; Secondary 05A15, 15A18
Published electronically: July 25, 2002
MathSciNet review: 1926836
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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.


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Additional Information

J. Marshall Ash
Affiliation: Mathematics Department, DePaul University, Chicago, Illinois 60614
Email: mash@math.depaul.edu

Gang Wang
Affiliation: Mathematics Department, DePaul University, Chicago, Illinois 60614
Email: gwang@math.depaul.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03086-6
PII: S 0002-9947(02)03086-6
Keywords: Abel summation, Baire category, Fourier series, generalized Laplacian, Green's function, $H^{J}$ sets, multiple trigonometric series, set of uniqueness, spherical convergence, subharmonic function, uniqueness
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
Article copyright: © Copyright 2002 American Mathematical Society