Eigenfunction expansions of ultradifferentiable functions and ultradistributions in \(\mathbb R^n\). (English) Zbl 1384.46030
The aim of the paper is to extend the results by T. Gramchev et al. [Proc. Am. Math. Soc. 139, No. 12, 4361–4368 (2011; Zbl 1231.35133)] by supplying characterizations of the Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for Fourier coefficients with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. (Shubin-type differential operators are \(m\)-order differential operators with polynomial coefficients.) Further, the authors prove that these eigenfunctions are absolute Schauder bases for the above spaces of ultradifferentiable functions. The paper generalizes to \(\mathbb{R}^n\) the results of A. Dasgupta and M. Ruzhansky [Trans. Am. Math. Soc. 368, No. 12, 8481–8498 (2016; Zbl 1366.46024)] with respect to compact manifolds.
Reviewer: László Simon (Budapest)
MSC:
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
Keywords:
Shubin-type differential operators; eigenfunction expansions; Gelfand-Shilov spaces; ultradifferentiable functions; ultradistributions; Denjoy-Carleman classesReferences:
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