Some spherical uniqueness theorems for multiple trigonometric series

J.Marshall Ash and Gang Wang

Review 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 $$\sum_{R/2\le|k|<R}|a_k|^2=o(R^2)\quad\text{as} \ R\to\infty;\tag 2$$ 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 Cantor-Lebesgue 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

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