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On the support of tempered distributions

Part of: Harmonic analysis in several variables Distributions, generalized functions, distribution spaces

Published online by Cambridge University Press:  12 January 2010

Jasson Vindas  and
Ricardo Estrada
Jasson Vindas
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA, Email: (jvindas@math.lsu.edu, restrada@math.lsu.edu)
Ricardo Estrada
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA, Email: (jvindas@math.lsu.edu, restrada@math.lsu.edu)
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Abstract

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We show that if the summability means in the Fourier inversion formula for a tempered distribution f ∈ S′(ℝn) converge to zero pointwise in an open set Ω, and if those means are locally bounded in L1(Ω), then Ω ⊂ ℝn\supp f. We prove this for several summability procedures, in particular for Abel summability, Cesàro summability and Gauss-Weierstrass summability.

Keywords

support of tempered distributionsFourier transformsCesàro summabilityAbel summabilityspherical means

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 46F10: Operations with distributions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 1 , February 2010 , pp. 255 - 270
Copyright
Copyright © Edinburgh Mathematical Society 2010