## 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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