A Top-Down Account of Linear Canonical Transforms
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 introdu...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2012 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2012
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148472 |
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| Zitieren: | 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. |
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Wolf, K.B. 2019-02-18T13:29:59Z 2019-02-18T13:29:59Z 2012 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. 1815-0659 2010 Mathematics Subject Classification: 20C10; 20C35; 33C15; 33C45 DOI: http://dx.doi.org/10.3842/SIGMA.2012.033 https://nasplib.isofts.kiev.ua/handle/123456789/148472 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. 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Top-Down Account of Linear Canonical Transforms Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Top-Down Account of Linear Canonical Transforms |
| spellingShingle |
A Top-Down Account of Linear Canonical Transforms Wolf, K.B. |
| title_short |
A Top-Down Account of Linear Canonical Transforms |
| title_full |
A Top-Down Account of Linear Canonical Transforms |
| title_fullStr |
A Top-Down Account of Linear Canonical Transforms |
| title_full_unstemmed |
A Top-Down Account of Linear Canonical Transforms |
| title_sort |
top-down account of linear canonical transforms |
| author |
Wolf, K.B. |
| author_facet |
Wolf, K.B. |
| publishDate |
2012 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148472 |
| citation_txt |
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. |
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| first_indexed |
2025-12-07T15:20:05Z |
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2025-12-07T15:20:05Z |
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