Some spherical uniqueness theorems for multiple trigonometric series

Details Some spherical uniqueness theorems for multiple trigonometric series Some spherical uniqueness theorems for multiple trigonometric series

2000-08-03

arxiv.org/abs/math/0008031v1

Ann. of Math. (2) 151 (2000), no. 1, 1--33

Mathematics

Classical Analysis and ODEs

33 pages

Scientific paper

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.

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