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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenfunction expansions in $ {\mathbb{R}}^n$


Authors: Todor Gramchev, Stevan Pilipovic and Luigi Rodino
Journal: Proc. Amer. Math. Soc. 139 (2011), 4361-4368
MSC (2010): Primary 35S05; Secondary 46F05, 35B65
Published electronically: April 5, 2011
MathSciNet review: 2823081
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Abstract: The main goal of this paper is to extend in $ \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 $ \mathbb{R}^n$ a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues $ \lambda_j, {j\in\mathbb{N}},$ and a corresponding sequence of eigenfunctions $ u_j, j\in \mathbb{N}$, forming an orthonormal basis of $ L^2(\mathbb{R}^n).$ Elements of Schwartz $ \mathcal S(\mathbb{R}^n)$, resp. Gelfand-Shilov $ S^{1/2}_{1/2}$ spaces, are characterized through expansions $ \sum_ja_ju_j$ and the estimates of coefficients $ a_j$ by the power function, resp. exponential function of $ \lambda_j$.


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Additional Information

Todor Gramchev
Affiliation: Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Email: todor@unica.it

Stevan Pilipovic
Affiliation: Institute of Mathematics, University of Novi Sad, trg. D. Obradovica 4, 21000 Novi Sad, Serbia
Email: stevan.pilipovic@uns.dmi.ac.rs

Luigi Rodino
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Email: luigi.rodino@unito.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11022-0
PII: S 0002-9939(2011)11022-0
Keywords: Shubin-type operators, Gelfand–Shilov spaces
Received by editor(s): October 11, 2010
Published electronically: April 5, 2011
Additional Notes: The first author was partially supported by a PRIN project of MIUR, Italy and GNAMPA, INDAM
The second author was supported by the project 144016, Serbia
The third author was partially supported by a PRIN project of MIUR, Italy and GNAMPA, INDAM
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society