Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $\mathbb R^n$

TL;DR

This paper characterizes ultradifferentiable function spaces on ^n using eigenfunction expansions of elliptic operators, showing these eigenfunctions form a basis and extending previous results to broader classes and ^n.

Contribution

It provides a new characterization of Gelfand-Shilov spaces via eigenfunction decay estimates, generalizing earlier work from Gevrey classes and compact manifolds to ^n.

Findings

Eigenfunctions form absolute Schauder bases for ultradifferentiable spaces.

Characterization extends previous results to ^n and broader ultradifferentiable classes.

Eigenfunction decay estimates precisely describe the function spaces.

Abstract

We obtain a characterization of ${\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb R^n)$ and $\mathcal {S}^{(M_p)}_{(M_p)}(\mathbb {R}^n)$, the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier results by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361-4368) for Gevrey weight sequences. It also generalizes to $\mathbb{R}^{n}$ recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds.