Uniqueness of square convergent trigonometric series. (English) Zbl 0840.42014

Ha, Young-Hwa; Lee, Jin

J. Korean Math. Soc. 32, No. 4, 785-802 (1995).

The authors introduce the notions of square like partial sum and convergence of multiple trigonometric series, and give a partial answer to the uniqueness problem for square convergence. Instead of going into details, we present here a corollary of their main result.
If the double trigonometric series \[ \sum_{m,n\in Z}\sum a_{mn} e^{i(mx+ ny)} \] is such that \[ \sum_{|m|< r}\sum_{|n|< r} a_{mn} e^{i(mx+ ny)}= o(1/r)\text{ and } \sum_{\max\{|m|, |n|\}= r} |a_{mn}|= o(1/r)\text{ as }r\to \infty, \] then \(a_{mn}= 0\) for all \(m,n\in Z\).

Reviewer: F.Móricz (Szeged)

MSC:

42B99	Harmonic analysis in several variables
42A63	Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization

Keywords:

square like partial sum; multiple trigonometric series; uniqueness; square convergence