EIGENFUNCTION EXPANSIONS IN Rn.

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 ϵ N, and a corresponding sequence of eigenfunctions uj, j ϵ N, forming an orthonormal basis of L²(Rn). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S½½ spaces, are characterized through expansions Σj ajuj and the estimates of coefficients aj by the power function, resp. exponential function of λj . [ABSTRACT FROM AUTHOR]