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A Top-Down Account of Linear Canonical Transforms
Репозиторій DSpace/Manakin
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2012, том 8
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A Top-Down Account of Linear Canonical Transforms
Інший ID:
2010 Mathematics Subject Classification: 20C10; 20C35; 33C15; 33C45
DOI: http://dx.doi.org/10.3842/SIGMA.2012.033
DOI: http://dx.doi.org/10.3842/SIGMA.2012.033
Посилання:
A Top-Down Account of Linear Canonical Transforms / K.B. Wolf // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 31 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html.
The Symposium on Superintegrability, Exact Solvability and Special Functions (Cuernavaca, 20–24 February 2012) was supported by the Centro Internacional de Ciencias AC, Fondo “Alfonso N´apoles G´andara”, Instituto de Ciencias F´ısicas, Instituto de Matem´aticas, and Intercambio Acad´emico of the Coordinaci´on de la Investigaci´on Cient´ıfica, Universidad Nacional Aut´onoma de M´exico. This work was supported by the Optica Matem´atica ´ projects PAPIIT-UNAM 101011 and SEP-CONACyT 79899.
Дата:
2012
Переглядів:
608
Завантажень:
243
Анотація:
We contend that what are called Linear Canonical Transforms (LCTs) should be seen as a part of the theory of unitary irreducible representations of the '2+1' Lorentz group. The integral kernel representation found by Collins, Moshinsky and Quesne, and the radial and hyperbolic LCTs introduced thereafter, belong to the discrete and continuous representation series of the Lorentz group in its parabolic subgroup reduction. The reduction by the elliptic and hyperbolic subgroups can also be considered to yield LCTs that act on functions, discrete or continuous in other Hilbert spaces. We gather the summation and integration kernels reported by Basu and Wolf when studiying all discrete, continuous, and mixed representations of the linear group of 2×2 real matrices. We add some comments on why all should be considered canonical.
