The Trouv\'e group for spaces of test functions

TL;DR

This paper characterizes the Trouvée group for various regularity classes as a natural Lie group of diffeomorphisms, establishing its structure and the continuity of the flow mapping, with applications to Bergman spaces.

Contribution

It proves that the Trouvée group coincides with a connected component of diffeomorphisms of certain regularity classes and endows it with a natural regular Lie group structure.

Findings

Trouvée group equals the connected component of certain diffeomorphisms.

The flow map from vector fields to diffeomorphisms is continuous.

Bergman spaces on polystrips are stable under ODE solutions.

Abstract

The Trouv\'e group $\mathcal G_{\mathcal A}$ from image analysis consists of the flows at a fixed time of all time-dependent vectors fields of a given regularity $\mathcal A(\mathbb R^d,\mathbb R^d)$. For a multitude of regularity classes $\mathcal A$, we prove that the Trouv\'e group $\mathcal G_{\mathcal A}$ coincides with the connected component of the identity of the group of orientation preserving diffeomorphims of $\mathbb R^d$ which differ from the identity by a mapping of class $\mathcal A$. We thus conclude that $\mathcal G_{\mathcal A}$ has a natural regular Lie group structure. In many cases we show that the mapping which takes a time-dependent vector field to its flow is continuous. As a consequence we obtain that the scale of Bergman spaces on the polystrip with variable width is stable under solving ordinary differential equations.