Hermite Functions, Lie Groups and Fourier Analysis

In this paper, we present recent results in harmonic analysis in the real line R and in the half-line R + , which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done in terms of a unified framework based on the use...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Switzerland), 2018-10, Vol.20 (11), p.816
Main Authors: Celeghini, Enrico, Gadella, Manuel, Del Olmo, Mariano A
Format: Article
Language:English
Subjects:
Algebra
Decomposition
Fourier analysis
Fourier transforms
Group theory
Harmonic analysis
Hilbert space
Laguerre functions
Lie groups
Mathematical functions
Quantum mechanics
Quantum physics
rigged Hilbert spaces
Signal processing
special functions
Symmetry
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Summary:In this paper, we present recent results in harmonic analysis in the real line R and in the half-line R + , which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done in terms of a unified framework based on the use of rigged Hilbert spaces. We find a relation between the universal enveloping algebra of the symmetry groups with the fractional Fourier transform. The results obtained are relevant in quantum mechanics as well as in signal processing as Fourier analysis has a close relation with signal filters. In addition, we introduce some new results concerning a discretized Fourier transform on the circle. We introduce new functions on the circle constructed with the use of Hermite functions with interesting properties under Fourier transformations.
ISSN:1099-4300
1099-4300
DOI:10.3390/e20110816