Uniqueness of rectangularly convergent trigonometric series
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.
scientific article; zbMATH DE number 221041