A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

On an inversion formula for the fourier transform on distributions by means of Gaussian functions

Gaussian functions are useful in order to establish inversion formulae for the classical Fourier transform. In this paper we show that they also are helpful in order to obtain a Fourier inversion formula for the distributional case.

On the support of tempered distributions

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.