A Fréchet Lie group on distributions

Abstract

Solving non-autonomous systems of ordinary differential equations leads to consider a new product of bivariate distributions called the $\star$~product in the literature. This product, distinct from the convolution product, has recently been used to establish structural results concerning non-autonomous differential systems, yet its formal underpinnings remain unclear. We demonstrate that it is well-defined on the weak closure of the space of smooth functions on a compact subset of ${\mathbb{R}}^2$. We establish that a subset of this weak closure has the structure of a Fréchet space $\mathcal{D}$. The $\star$~product arises from the composition of endomorphisms of that space. Invertible elements of $\mathcal{D}$ form a dense subset of it and a Fréchet Lie group for the operation $\star$. This product generalizes the convolution, Volterra compositions of first and second type and induces Schwartz's bracket.