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### Eigenfunction expansions of ultradifferentiable functions and ultradistributions in ℝⁿ

Dorde Vuckovic UGent and Jasson Vindas Diaz UGent (2016) 7(4). p.519-531
abstract
We obtain a characterization of ${\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb{R}^n)$ and $\mathcal {S}^{(M_p)}_{(M_p)}(\mathbb{R}^n)$, the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier results by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361--4368) for Gevrey weight sequences. It also generalizes to $\mathbb{R}^{n}$ recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds.
Please use this url to cite or link to this publication:
author
organization
year
type
journalArticle (original)
publication status
published
subject
keyword
eigenfunction expansions, Shubin type differential operators, Gelfand-Shilov spaces, ultradifferentiable functions, ultradistributions, Denjoy-Carleman classes, SPACES
journal title
JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS
J. Pseudo-Differ. Oper. Appl.
volume
7
issue
4
pages
519 - 531
Web of Science type
Article
Web of Science id
000386622500005
JCR category
MATHEMATICS
JCR impact factor
0.529 (2016)
JCR rank
199/310 (2016)
JCR quartile
3 (2016)
ISSN
1662-9981
DOI
10.1007/s11868-016-0157-9
project
Quasianalytic and non-quasianalytic classes in Fourier analysis and approximation theory
language
English
UGent publication?
yes
classification
A1
copyright statement
I have transferred the copyright for this publication to the publisher
id
8159733
handle
http://hdl.handle.net/1854/LU-8159733
date created
2016-11-20 13:49:14
date last changed
2018-01-03 23:30:24
@article{8159733,
abstract     = {We obtain a characterization of \$\{{\textbackslash}mathcal S\}\^{ }\{{\textbackslash}\{M\_p{\textbackslash}\}\}\_\{{\textbackslash}\{M\_p{\textbackslash}\}\}({\textbackslash}mathbb\{R\}\^{ }n)\$ and \${\textbackslash}mathcal \{S\}\^{ }\{(M\_p)\}\_\{(M\_p)\}({\textbackslash}mathbb\{R\}\^{ }n)\$, the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier results by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361--4368) for Gevrey weight sequences. It also generalizes to \${\textbackslash}mathbb\{R\}\^{ }\{n\}\$ recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds.},
author       = {Vuckovic, Dorde and Vindas Diaz, Jasson},
issn         = {1662-9981},
journal      = {JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS},
keyword      = {eigenfunction expansions,Shubin type differential operators,Gelfand-Shilov spaces,ultradifferentiable functions,ultradistributions,Denjoy-Carleman classes,SPACES},
language     = {eng},
number       = {4},
pages        = {519--531},
title        = {Eigenfunction expansions of ultradifferentiable functions and ultradistributions in \unmatched{211d}\unmatched{207f}},
url          = {http://dx.doi.org/10.1007/s11868-016-0157-9},
volume       = {7},
year         = {2016},
}


Chicago
Vuckovic, Dorde, and Jasson Vindas Diaz. 2016. “Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions in ℝn.” Journal of Pseudo-differential Operators and Applications 7 (4): 519–531.
APA
Vuckovic, D., & Vindas Diaz, J. (2016). Eigenfunction expansions of ultradifferentiable functions and ultradistributions in ℝn. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, 7(4), 519–531.
Vancouver
1.
Vuckovic D, Vindas Diaz J. Eigenfunction expansions of ultradifferentiable functions and ultradistributions in ℝn. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS. 2016;7(4):519–31.
MLA
Vuckovic, Dorde, and Jasson Vindas Diaz. “Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions in ℝn.” JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS 7.4 (2016): 519–531. Print.