Hermite expansions and products of tempered distributions

Abstract

The space of tempered distributions 𝒮′ can be realized as a sequence spaces by means of the Hermite representation theorem. In this article, we introduce and study two new products of tempered distributions based on this Hermite representation theorem. In particular, we obtain the products [H]δ=δ/2, [δ] vp(1/x)=−δ′ and [δ(r)]vp(1/x)=−(δ(r+1))/(r+1) for even r.