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

Nuno Costa Dias; Joao Nuno Prata

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arXiv · 0907.2069

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

Nuno Costa Dias·Joao Nuno Prata

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Abstract

We construct a generalization of the multiplicative product of distributions presented by L. Hörmander in [L. Hörmander, {\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 $ψ$ into another, ordinary linear differential equation, with general solution $χ_Ω ψ$, where $χ_Ω$ is the characteristic function of some prescribed interval $Ω\subset \bkR$.

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Nuno Costa Dias, Joao Nuno Prata. 2009-07-12. A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients. https://arxiv.org/abs/0907.2069

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