Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations

TL;DR

This paper investigates the structure of coefficient spaces in eigenfunction expansions of ultradifferentiable functions on compact manifolds, revealing their perfect sequence space nature and tensor representations for linear mappings.

Contribution

It extends previous work by characterizing the tensor structure and adjoint mappings of Fourier coefficient spaces in Komatsu classes, including analytic and Gevrey functions.

Findings

Fourier coefficient spaces are perfect sequence spaces

Tensor structures of sequential mappings are described

Linear mappings between ultradifferentiable spaces have tensor representations

Abstract

In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mappings between these spaces on compact manifolds.