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

DOI:
https://doi.org/10.1090/S0002-9947-02-03086-6

Published electronically:
July 25, 2002

MathSciNet review:
1926836

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Abstract: A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is 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:
https://doi.org/10.1090/S0002-9947-02-03086-6

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