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--- abstract: '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 ${\mathcal A}(\bkR)$ 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 $\Omega \subset \bkR$.' address: 'Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias, Av. Campo Grande 376, 1749-024 Lisboa, Portugal and Grupo de Física Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal' author: - Nuno Costa Dias and João Nuno Prata title: A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients --- [^1] Introduction ============ Two of the most interesting properties of the space of Schwartz distributions ${\mathcal D}'(\bkR^n)$ over $\bkR^n$ are that the space of continuous functions $C^{0}(\bkR^n)$ is a subspace of ${\mathcal D}'(\bkR^n)$ and that differentiation is an internal operation in ${\mathcal D}'(\bkR^n)$ [@Schwartz1; @Zemanian; @Hormander]. On the other hand, its major limitation is that it only displays the structure of a vector space and not that of an algebra. This has been known since 1954 when L. Schwartz proved that there is [*no*]{} associative and commutative algebra of generalized functions $({\mathcal A}(\bkR^n),+,\circ)$ satisfying the three following properties:\ \ (1) The space of distributions ${\mathcal D}'(\bkR^n)$ over $\bkR^n$ is linearly embedded into ${\mathcal A}(\bkR^n)$ and $f(x)\equiv 1$ is the identity in ${\mathcal A}(\bkR^n)$.\ \ (2) The restriction of the product $\circ$ to the set of continuous functions $C^0(\bkR^n)$ reproduces the pointwise product of functions.\ \ (3) There exist linear derivative operators $\frac{\partial}{\partial {x_i}}:{\mathcal A}(\bkR^n)\to {\mathcal A}(\bkR^n)$, $(i=1,..,n)$ that: (a) Satisfy the Leibniz rule and (b) Their restrictions to ${\mathcal D}'(\bkR^n)$ coincide with the usual distributional derivatives.\ This theorem became known as the Schwartz impossibility result [@Schwartz2]. In spite of it, certain products of distributions emerge rather naturally in several fields of research in both mathematics and physics. Common examples can be found in quantum electrodynamics and particle physics when dealing with self interacting terms; in hydrodynamics in the formulation of the dynamics of shock waves; and, more generally, whenever one wants to consider differential equations where either the coefficients or the prospective solutions are non-smooth. These issues motivated several proposals towards a precise construction of a multiplicative product of distributions. Among these, the most famous example is due to J. Colombeau [@Colombeau0; @Colombeau; @Colombeau2], who constructed differential algebras of generalized functions ${\mathcal G}(\bkR^n)$ satisfying conditions (1) and (3) with (2) holding only in $C^{\infty}(\bkR^n)$ (another