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 $\mathcal{A}(\text{bkR})$ of piecewise smooth functions $f:\text{bkR}\to \text{bkC}$ and all their (distributional) derivatives. It is associative, satisfies the Leibniz rule and reproduces the usual pointwise product of functions for regular distributions in $\mathcal{A}(\text{bkR})$. Endowed with this product, the space $\mathcal{A}(\text{bkR})$ 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 ${\chi }_{\mathrm{\Omega }}\psi$, where ${\chi }_{\mathrm{\Omega }}$ is the characteristic function of some prescribed interval $\mathrm{\Omega }\subset \text{bkR}$.