A characterization of the generalized functions via the special Hermite expansions. (English) Zbl 1345.46034
The authors give the correspondence between the Gel’fand-Shilov space \(\mathcal S_r^r(\mathbb R^{2d})\) and a sequence with exponential decay and the dual space of the Gel’fand-Shilov space \((\mathcal S_r^r)'(\mathbb R^{2d})\) and a sequence space with exponential growth by means of the special Hermite function introduced by R. S. Strichartz [J. Funct. Anal. 87, No. 1, 51–148 (1989; Zbl 0694.43008)]. However, this result was already obtained by G.-Z. Zhang [Chinese Math. 4, 211–221 (1963; Zbl 0194.15001); translation from Acta Math. Sin. 13, 193–204 (1963)] for all \(d\) instead of \(2d\) by means of the Hermite functions as the authors mention in the introduction.
Reviewer: Dohan Kim (Seoul)
MSC:
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 46F12 | Integral transforms in distribution spaces |
| 46F15 | Hyperfunctions, analytic functionals |
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
References:
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