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 χΩψ, where χΩ is the characteristic function of some prescribed interval Ω ⊂ R.