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Keywords = bases in Hilbert space L2(ℝ)

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21 pages, 363 KiB  
Article
Heisenberg–Weyl Groups and Generalized Hermite Functions
by Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo
Symmetry 2021, 13(6), 1060; https://doi.org/10.3390/sym13061060 - 12 Jun 2021
Cited by 4 | Viewed by 2751
Abstract
We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions [...] Read more.
We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Heisenberg–Weyl group and some of their extensions. Full article
(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)
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