×

On Fourier transforms of functions supported on sets of finite Lebesgue measure. (English) Zbl 0576.42016

Let G be a locally compact Abelian group with dual group \(\hat G.\) Denote the Haar measure on G and \(\hat G\) by m and \(\hat m,\) respectively. T. Matolcsi and J. Szücs [C. R. Acad. Sci., Paris, Sér. A 277, 841-843 (1973; Zbl 0266.43002)] have proved that if \(f\in L^ 1(G)\) and \[ (*)\quad m\{x\in G:f(x)\neq 0\}\cdot \hat m\{\gamma \in \hat G:\hat f(\gamma)\neq 0\}<1, \] then \(f=0\) a.e. [m], where \(\hat f\) denotes the Fourier transform of f. In this note the author proves that if \(G={\mathbb{R}}^ n\), then the condition (*) in the Matolcsi-Szücs result can be replaced by \(m\{x\in {\mathbb{R}}^ n:f(x)\neq 0\}<\infty\) and \(\hat m\{\) \(\gamma\in {\mathbb{R}}^ n:\hat f(\gamma)\neq 0\}<\infty\).
Reviewer: L.Yap

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Citations:

Zbl 0266.43002
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amrein, W. O.; Berthier, A. M., On support properties of \(L^p\)-functions and their Fourier transforms, J. Funct. Anal., 24, 258-267 (1977) · Zbl 0355.42015
[2] M. BenedicksMath. Scand.; M. BenedicksMath. Scand. · Zbl 0442.42007
[3] Matolcsi, T.; Szücs, J., Intersection des mesures spectrales conjugées, C. R. Acad. Sci. Paris Sér. A, 277, 841-843 (1973) · Zbl 0266.43002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.