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. Hörmander in [L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 1983]. The new product is defined in the vector space A ( R ) of piecewise smooth functions f : R → C and all their (distributional) derivatives. It is associative, satisfies the Leibniz rule and reproduces the usual pointwise product of functions for regular distributions in A ( R ) . Endowed with this product, the space A ( R ) becomes a differential associative algebra of generalized functions. By working in the new A ( R ) -setting we determine a method for transforming an ordinary linear differential equation with general solution ψ into another, ordinary linear differential equation, with general solution χ Ω ψ , where χ Ω is the characteristic function of some prescribed interval Ω ⊂ R .