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

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• 作者：Đorđe Vučković ; Jasson Vindas
• 关键词：Shubin type differential operators ; Eigenfunction expansions ; Gelfand–Shilov spaces ; Ultradifferentiable functions ; Ultradistributions ; Denjoy–Carleman classes
• 刊名：Journal of Pseudo-Differential Operators and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：7
• 期：4
• 页码：519-531
• 全文大小：486 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：None Assigned
• 出版者：Birkh盲user Basel
• ISSN：1662-999X
• 卷排序：7

文摘

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 Am Math Soc 139:4361–4368, 2011) 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.