Eigenfunction expansions in ${\mathbb R}^n$
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- by Todor Gramchev, Stevan Pilipovic and Luigi Rodino
- Proc. Amer. Math. Soc. 139 (2011), 4361-4368
- DOI: https://doi.org/10.1090/S0002-9939-2011-11022-0
- Published electronically: April 5, 2011
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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 $\lambda _j$.References
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Bibliographic 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
- MR Author ID: 149460
- Email: luigi.rodino@unito.it
- 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
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 4361-4368
- MSC (2010): Primary 35S05; Secondary 46F05, 35B65
- DOI: https://doi.org/10.1090/S0002-9939-2011-11022-0
- MathSciNet review: 2823081