A note on a family of distributional products important in the applications

Abstract

We define a family of products of a distribution $T'∈ D'$ by a distribution $S∈ C^∈fty ⨁ D'_n$ where $D'_n$ means the space of distributions with support nowhere dense. Each product depends on the choice of a group G of unimodular transformations and a function $?∈ D$ with $∈t ?=1$ which is G-invariant.These products are consistent with the usual product of a distribution by a $C^∈fty$- function, their outcome distributive, and verify also the usual law of the derivate of a product together with being invariant by translation and all transformations in G. A sufficient condition for associativity is given. Simple physical interpretations of the products $Hδ$ and $δδ$, where H is the Heaviside function and δ is the Dirac’s measure, are considered. In particular we discuss certain shock wave solution of the differential equation $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$$.