Detection of signals with the orthogonal basis of the Gauss–Hermite functions

2018 ◽  
Vol 1 (1) ◽  
pp. 48-61
Author(s):  
D. A. Balakin ◽  
◽  
S. S. Churkin ◽  
V. V. Shtykov ◽  
◽  
...  
Keyword(s):  
Orthogonal Basis ◽  
Hermite Functions ◽  
Detection Of Signals

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.

Properties of the representation of the Fokker-Planck equation in the basis of Hermite functions

2004 ◽  
Vol 49 (3) ◽  
pp. 139-142
Author(s):  
P. A. Zakharchenko ◽  
E. V. Radkevich
Keyword(s):  
Planck Equation ◽  
Hermite Functions ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

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