On the support of tempered distributions

We show that if the summability means in the Fourier inversion formula for a tempered distribution f is an element of S'(R-n) converge to zero pointwise in an open set Omega, and if those means are locally bounded in L-1 (Omega), then Omega subset of R-n\supp f. We prove this for several summability procedures, in particular for Abel summability, Cesaro summability and Gauss-Weierstrass summability