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Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 128, Number 9, Pages 2627-2636
S 0002-9939(00)05272-2
Article electronically published on February 25, 2000
NEW UNIQUENESS THEOREMS
FOR TRIGONOMETRIC SERIES
J. MARSHALL ASH AND SH. T. TETUNASHVILI
(Communicated by Christopher D. Sogge)
ABSTRACT. A uniqueness theorem is proved for trigonometric series and an-
other 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 2d 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."
1. RESULTS
1.1. One dimensional results.
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