Qualitative uncertainty principles for the generalized Fourier transform associated to a Dunkl type operator on the real line

2015 ◽  
Vol 6 (2) ◽  
pp. 141-162 ◽  
Author(s):  
Hatem Mejjaoli ◽  
Khalifa Trimèche
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Type Operator ◽  
Uncertainty Principles ◽  
Generalized Fourier Transform ◽  
The Real ◽  
Qualitative Uncertainty

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

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 Weyl–Heisenberg group and some of their extensions.

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