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

Abstract

We construct a generalization of the multiplicative product of distributions presented by L. Hörmander in [L. Hörmander, {\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 $Ω\subset \bkR$.

23 pages, Latex file

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