Eigenfunction expansions in ${\mathbb{R}}^n$

The main goal of this paper is to extend in $\mathbb{R}^n$ a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in $\mathbb{R}^n$ a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues $\lambda_j, {j\in\mathbb{N}},$ and a corresponding sequence of eigenfunctions $u_j, j\in \mathbb{N}$, forming an orthonormal basis of $L^2(\mathbb{R}^n).$ Elements of Schwartz $\mathcal S(\mathbb{R}^n)$, resp. Gelfand-Shilov $S^{1/2}_{1/2}$ spaces, are characterized through expansions $\sum_ja_ju_j$ and the estimates of coefficients $a_j$ by the power function, resp. exponential function of $\lambda_j$.