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.


42B99 Harmonic analysis in several variables
40B05 Multiple sequences and series
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