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Some spherical uniqueness theorems for multiple trigonometric series. (English) Zbl 0955.42010

The following uniqueness theorem is proved for the multiple trigonometric series \[ \sum_{k\in\mathbb Z^n}a_k e^{ikx},\tag{1} \] 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 \[ \sum_{R/2\leq|k|<R}|a_k|^2=o(R^2)\quad\text{as} \;R\to\infty;\tag{2} \] 2. \(f^*(x)\) and \(f_*(x),\) \(\limsup\) and \(\liminf\) of the Abel means of (1), respectively, are finite for all \(x;\) 3. \(\min\{\text{Re}f_*(x), \text{Im}f_*(x)\}\geq A(x)\in L^1(\mathbb T^n).\) Then \(f_*\in L^1(\mathbb T^n)\) 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.

MSC:

42B05 Fourier series and coefficients in several variables
42B08 Summability in several variables
42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
42B15 Multipliers for harmonic analysis in several variables
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