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

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
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