Some spherical uniqueness theorems for multiple trigonometric series
Abstract
We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function $f(x)$ which is bounded below by an integrable function, then the series is the Fourier series of $f(x)$ if the coefficients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically convergent everywhere to an everywhere finite integrable function $f(x)$, then the series is the Fourier series of $f(x)$. We also show that a singleton is a set of uniqueness. These results are generalizations of a recent theorem of J. Bourgain and some results of V. Shapiro.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2000
 arXiv:
 arXiv:math/0008031
 Bibcode:
 2000math......8031A
 Keywords:

 Classical Analysis and ODEs;
 42B05;
 42B99
 EPrint:
 33 pages