Eigenfunction expansions in ℝⁿ

Abstract

Abstract. 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 λ j , j ∈ N, and a corresponding sequence of eigenfunctions u j , j ∈ N, forming an orthonormal basis of L 2 (R n ).

Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $$\mathbb R^n$$ R n

“…, our result includes as particular instances those from [9]. In the special case of the harmonic oscillator…”

“…In this section we exploit the iterative approach from [2,9,18] in order to obtain a structural characterization of S * (R n ) in terms of the growth of the L 2 norms of the iterates of the operator P . The regularity result Theorem 1.2 will readily follow from Theorem 3.4 below.…”

“…Recall [17,19] that global ellipticity means that the principal symbol Our starting point is the same as in [9], i.e., the interpolating inequality [9, Prop.…”

“…In [9] Gramchev, Pilipović, and Rodino used this type of operators to give an analogue to Seeley's result for some classes of Gelfand-Shilov spaces. The aim of this paper is to extend the results from [9] by supplying a characterization of the general Gelfand-Shilov spaces S {Mp} (R n ) = S {Mp} {Mp} (R n ) and…”

“…The aim of this paper is to extend the results from [9] by supplying a characterization of the general Gelfand-Shilov spaces S {Mp} (R n ) = S {Mp} {Mp} (R n ) and…”