A Fréchet Lie group on distributions
Manon Ryckebusch, Abderrahman Bouhamidi, Pierre-Louis Giscard
TL;DR
The paper introduces a novel star_I product on bivariate distributions, defined as $(f\star_I g)(x,y)=\int_I f(x,\tau) g(\tau,y) \,d\tau$, and places it in a Mikusiński-style framework by working on the weak closure of $\mathcal{C}^{\infty}(I^2)$. It shows the product is well-defined, associative, and extends to a set $\mathcal{D}$ of distribution-like objects $d(x,y)=\tilde{d}(x,y)\Theta(x-y)+\sum_i \tilde{d}_i(x,y)\delta^{(i)}(x-y)$ with $\delta$ as the unit. The authors relate $\star_I$ to existing products (convolution, Volterra compositions, and pointwise products) and interpret the product as composition of endomorphisms, enabling a transfer to linear functionals and the Schwartz bracket in higher dimensions. The core contribution is constructing Inv$(\mathcal{D})$, the dense set of star-invertible elements, and proving it forms a Fréchet Lie group under $\star$, acting on $\mathcal{C}^{\infty}(I^2)$ and embedding into $\mathrm{Aut}(\mathcal{C}^{\infty}(I^2))$. This framework unifies several integral-operator products and provides a rigorous foundation for using $\star$-products in non-autonomous differential systems and related physics applications.
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.