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Annals of Mathematics, II. Series Vol. 151, No. 1, pp. 133 (2000) 

Some spherical uniqueness theorems for multiple trigonometric seriesJ.Marshall Ash and Gang WangReview from Zentralblatt MATH: The following uniqueness theorem is proved for the multiple trigonometric series $$\sum_{k\in\Bbb Z^n}a_k e^{ikx},\tag 1$$ where the coefficients $a_k$ are arbitrary complex numbers and $kx=k_1x_1+ \cdots+k_nx_n.$ Denote $k^2=k_1^2+\cdots+k_n^2.$ Theorem. Suppose that 1. the coefficients $a_k$ satisfy $$\sum_{R/2\lek<R}a_k^2=o(R^2)\quad\text{as} \ R\to\infty;\tag 2$$ 2. $f^*(x)$ and $f_*(x),$ $\limsup$ and $\liminf$ of the Abel means of (1), respectively, are finite for all $x;$ 3. $\min\{\text{Re}f_*(x), \text{Im}f_*(x)\}\ge A(x)\in L^1(\Bbb T^n).$ Then $f_*\in L^1(\Bbb T^n)$ and (1) is its Fourier series. This result and its consequences are generalizations of results of V. Shapiro and a recent result of J. Bourgain. Condition (2) is more general than that used by B. Connes for the multidimensional extension of the CantorLebesgue theorem. For several dimensions, this range of problems is far from being complete, and the paper under review is an important step forward. Reviewed by Elijah Liflyand Keywords: multiple trigonometric series; uniqueness; spherical Abel summability; spherical convergence; Laplacian; harmonic; subharmonic Classification (MSC2000): 42B05 42B08 42A63 42B15 Full text of the article: not available electronically
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
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