Application of the Rigged Hilbert Spaces into the Generalized Signals &amp;amp; Systems Theory

Heredia-Juesas, Juan; Gago-Ribas, E.; Vidal-Garcia, P.

doi:10.1109/iceaa.2015.7297341

Cited by 3 publications

(3 citation statements)

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“…In the last years the RHS have appeared associated to time-frequency analysis and Gabor analysis that have many applications in physics and engineering related to signal proceesing [43][44][45][46][47][48][49][50]. In particular, applications in electrical engineering have been introduced in [52][53][54].…”

Section: Rigged Hilbert Spacesmentioning

confidence: 99%

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Groups, Special Functions and Rigged Hilbert Spaces

Celeghini

Gadella

Olmo

2019

Axioms

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We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space, instead they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . As a matter of fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can be often continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We have investigated this formalism to various examples of interest in quantum mechanics. In particular, we have considered, SO(2) and functions on the unit circle, SU (2) and associated Laguerre functions, Weyl-Heisenberg group and Hermite functions, SO(3, 2) and spherical harmonics, su(1, 1) and Laguerre functions, su(2, 2) and algebraic Jacobi functions and, finally, su(1, 1)⊕su(1, 1) and Zernike functions on a circle.

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Section: Rigged Hilbert Spacesmentioning

confidence: 99%

“…The operators defined in (53) act on the functions Z m j (r, φ) exactly as K ± and K 3 on L m j (x), expressions given in (45). Also, we define the corresponding operators on H as…”

Section: Associated Laguerre Functions On the Planementioning

confidence: 99%

Groups, Special Functions and Rigged Hilbert Spaces

Celeghini

Gadella

Olmo

2019

Axioms

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“…Another application of RHS, which has a particular importance in our presentation, is signal processing. In particular, in the electrical engineering literature, these aspects have been discussed in [ 15 , 16 , 17 , 18 ].…”

Section: Introductionmentioning

confidence: 99%

Hermite Functions, Lie Groups and Fourier Analysis

Celeghini

Gadella

Olmo

2018

Entropy

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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 in 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. Also, 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.

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