EIGENFUNCTION EXPANSIONS IN Rn

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