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


We define a family of products of a distribution T'∈ D' by a distribution S∈ C^∈fty ⨁ D'<sub>n</sub> where D'<sub>n</sub> 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 𝛼∈ D with ∈t 𝛼=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

DOI Code: 10.1285/i15900932v7n2p151

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