A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients

TL;DR

This paper introduces a new associative, Leibniz-rule satisfying product of distributions within a piecewise smooth function space, enabling transformation of linear differential equations with distributional coefficients into equivalent forms.

Contribution

It generalizes Hörmander's product of distributions to a broader class of functions, forming a differential algebra and providing a method to transform differential equations with distributional coefficients.

Findings

Defined a new associative product of distributions in ${ mf A}(kR)$.

Established a differential algebra structure on ${ mf A}(kR)$.

Developed a method to transform differential equations using characteristic functions.

Abstract

We construct a generalization of the multiplicative product of distributions presented by L. H\"ormander in [L. H\"ormander, {\it The analysis of linear partial differential operators I} (Springer-Verlag, 1983)]. The new product is defined in the vector space ${\mathcal A}(\bkR)$ of piecewise smooth functions $f: \bkR \to \bkC$ and all their (distributional) derivatives. It is associative, satisfies the Leibniz rule and reproduces the usual pointwise product of functions for regular distributions in ${\mathcal A}(\bkR)$. Endowed with this product, the space ${\mathcal A}(\bkR)$ becomes a differential associative algebra of generalized functions. By working in the new ${\mathcal A}(\bkR)$-setting we determine a method for transforming an ordinary linear differential equation with general solution $\psi$ into another, ordinary linear differential equation, with general solution $\chi_{\Omega} \psi$, where $\chi_{\Omega}$ is the characteristic function of some prescribed interval $\Omega \subset \bkR$.