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

Abstract:

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 $\lambda _j$.