On the support of tempered distributions. (English) Zbl 1183.42011

Vindas, Jasson; Estrada, Ricardo

Proc. Edinb. Math. Soc., II. Ser. 53, No. 1, 255-270 (2010).

Summary: We show that if the summability means in the Fourier inversion formula for a tempered distribution \(f\in \mathcal S^{\prime}(\mathbb R^{n})\) converge to zero pointwise in an open set \(\varOmega \), and if those means are locally bounded in \(L^{1}(\varOmega)\), then \(\varOmega \subset \mathbb R^{n}\setminus \text{ supp } f\). We prove this for several summability procedures, in particular for Abel summability, Cesàro summability and Gauss-Weierstrass summability.

MSC:

42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F10	Operations with distributions and generalized functions
40C99	General summability methods

Keywords:

support of tempered distributions; Fourier transforms; Cesàro summability; Abel summability; spherical means