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Structural theorems for Gelfand–Shilov spaces. (English) Zbl 1171.46029

The authors give a structure theorem for Gelfand–Shilov spaces, \({\mathcal S}^{\{M_p\}^\prime }\) and \({\mathcal S}^{(M_p)^\prime }\), of Roumieu and Beurling type. The class of sequences of positive integers \(M_p\), \(p \in \mathbb{N}\), considered in this paper covers many spaces known in the literature. Several examples included in this framework are given. In order to obtain their structure theorem, the authors follow the ideas of B.Simon [J. Math.Phys.12, 140–148 (1971; Zbl 0205.12901)] and B.P.Dhungana and T.Matsuzawa [Res.Rep.Fac.Sci.Technol., Meijo Univ.45, 43–48 (2005; Zbl 1080.46022)] who used the characterization of Hermite coefficients of elements of the spaces to prove the structure theorems for the spaces of tempered distributions and Gevrey ultradistributions respectively. The main tool of the proof is the characterization of the Fourier–Hermite coefficients of Gelfand–Shilov spaces and their duals [M.Langenbruch, Manuscr.Math.119, No.3, 269–285 (2006; Zbl 1101.46026)], [Z.Lozanov–Crvenković and D.D.Perišić, Novi Sad J. Math.37, No.2, 129–147 (2007; Zbl 1274.46077)].

MSC:

46F05 Topological linear spaces of test functions, distributions and ultradistributions
46F12 Integral transforms in distribution spaces
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
35S99 Pseudodifferential operators and other generalizations of partial differential operators
46F10 Operations with distributions and generalized functions
46A45 Sequence spaces (including Köthe sequence spaces)
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