A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients

摘要

We construct a generalization of the multiplicative product of distributions presented by L. H?rmander in [L. H?rmander, 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.