1,843 research outputs found
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 into another, ordinary
linear differential equation, with general solution , where
is the characteristic function of some prescribed interval
\Omega \subset \bkR.Comment: 23 pages, Latex fil
- …