Inverse Closed Ultradifferential Subalgebras

TL;DR

This paper extends classical approximation theory to construct inverse-closed ultradifferentiable subalgebras, including Carleman classes and Dales-Davie algebras, with applications to matrices with exponential decay.

Contribution

It introduces a general framework for inverse-closed ultradifferentiable subalgebras, expanding previous work on smooth subalgebras to include Carleman classes and Dales-Davie algebras.

Findings

Established inverse-closedness for ultradifferentiable subalgebras.

Unified treatment of Carleman classes and Dales-Davie algebras.

Applied theory to matrices with exponential decay.

Abstract

In previous work we have shown that classical approximation theory provides methods for the systematic construction of inverse-closed smooth subalgebras. Now we extend this work to treat inverse-closed subalgebras of ultradifferentiable elements. In particular, Carleman classes and Dales-Davie algebras are treated. As an application the result of Demko, Smith and Moss and Jaffard on the inverse of a matrix with exponential decay is obtained within the framework of a general theory of smoothness.