Eigenfunction expansions in ℝⁿ

Abstract

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