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

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.Comment: 23 pages, Latex fil