Eigenfunction expansions in ℝⁿ

Abstract

The main goal of this paper is to extend in <!-- MATH $\mathbb{R}^n$ --> 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 <!-- MATH $\mathbb{R}^n$ --> a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues <!-- MATH $\lambda_j, {j\in\mathbb{N}},$ --> and a corresponding sequence of eigenfunctions <!-- MATH $u_j, j\in \mathbb{N}$ --> , forming an orthonormal basis of <!-- MATH $L^2(\mathbb{R}^n).$ --> Elements of Schwartz <!-- MATH $\mathcal S(\mathbb{R}^n)$ --> , resp. Gelfand-Shilov <!-- MATH $S^{1/2}_{1/2}$ --> spaces, are characterized through expansions <!-- MATH $\sum_ja_ju_j$ --> and the estimates of coefficients by the power function, resp. exponential function of .