Eigenfunction expansions of ultradifferentiable functions and ultradistributions

TL;DR

This paper provides a comprehensive global characterization of ultradifferentiable functions and ultradistributions on compact manifolds using eigenfunction expansions of elliptic operators, extending prior results for analytic and Gevrey classes.

Contribution

It extends the eigenfunction expansion characterization to general ultradifferentiable classes on compact manifolds, broadening the scope of previous work.

Findings

Eigenfunction expansions characterize ultradifferentiable functions on compact manifolds.

The results generalize earlier characterizations for analytic and Gevrey functions.

Provides a unified framework for ultradifferentiable classes on manifolds.

Abstract

In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold $X$. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on $X$. This extends the result for analytic functions on compact manifold by Seeley, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors.