  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
$\Omega \subset \bkR$.

Functional Analysis;Classical Analysis and ODEs