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Eigenfunction expansions of ultradifferentiable functions and ultradistributions in ℝⁿ

Dorde Vuckovic (UGent) and Jasson Vindas Diaz (UGent)
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Quasianalytic and non-quasianalytic classes in Fourier analysis and approximation theory
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.
Keywords
eigenfunction expansions, Shubin type differential operators, Gelfand-Shilov spaces, ultradifferentiable functions, ultradistributions, Denjoy-Carleman classes, SPACES

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Citation

Please use this url to cite or link to this publication:

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.
@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},
}

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