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Uniqueness of rectangularly convergent trigonometric series. (English) Zbl 0780.42015
For a vector $$X=(x_ 1,\ldots,x_ n)\in\mathbb{R}^ n$$ and (real or complex) coefficients $$a_ M$$, $$M=(m_ 1,\ldots,m_ n)\in\mathbb{Z}^ n$$, we consider the multiple trigonometric series $$T:=\sum_{M\in\mathbb{R}^ n}a_ Me^{iMX}$$. The type of convergence is crucial to the results of the paper: if $$Z=(z_ 1,\ldots,z_ n)\in\mathbb{Z}^ n$$ and we denote by $$R_ Z$$ the rectangle $$R_ Z:=\{M:| m_ j|\leq| z_ j|$$, $$j=1,\ldots,n\}$$, then $$T$$ is said to be rectangularly convergent when $$\exists\lim_{\min_{j=1,\ldots,n}| z_ j|\to\infty}\sum_{M\in R_ Z}a_ Me^{iMX}$$.
It was proved by Georg Cantor (1870) that if a one-dimensional trigonometric series converges to zero everywhere, then all of its coefficients must be zero. A discussion is given of various generalizations of Cantor’s result, of which the only nontrivial direct analogues are for double series, namely that if a double trigometric series is either rectangularly convergent, or circularly convergent, to zero everywhere, then all of its coefficients must be zero; however, even the corresponding theorem for square convergence of a double trigonometric series is still unknown, which indicates the delicacy of the problem.
The present paper deals decisively with the case of rectangular convergence by proving that if a multiple trigonometric series is rectangular convergent to zero at every point, then all of its coefficients are zero. The proof is a tour-de-force: the authors select those ideas of the masters (Cantor, Riemann, Schwarz,...) which are capable of generalization, but add their own substantial innovations. The bibliography is informative.

##### MSC:
 42B99 Harmonic analysis in several variables 40B05 Multiple sequences and series
##### Keywords:
multiple trigonometric series; rectangular convergence
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