Eigenfunction expansions in R^n

The main goal of this paper is to extend in Rn 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 Rn a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues λj, j ∈ N, and a corresponding sequence of eigenfunctions uj, j ∈ N, forming an orthonormal basis of L2(Rn). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S1/2 1/2 spaces, are characterized through expansions and the estimates of coefficients aj by the power function, resp. exponential function of λj .