Eigenfunction expansions of ultradifferentiable functions and ultradistributions in Rn

Abstract

We obtain a characterization of S{Mp}{Mp}(Rn) and S(Mp)(Mp)(Rn), 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 Am Math Soc 139:4361–4368, 2011) for Gevrey weight sequences. It also generalizes to Rn recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds.