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

Benedicks, Michael

J. Math. Anal. Appl. 106, 180-183 (1985).

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

Keywords:

locally compact Abelian group; Haar measure

Citations:

Zbl 0266.43002

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