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SIGMA, 2012, Volume 8, 033, 13 pages (Mi sigma710)  

This article is cited in 5 scientific papers (total in 5 papers)

A top-down account of linear canonical transforms

Kurt Bernardo Wolf

Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Mor. 62210, México

Abstract: 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 $2\times2$ real matrices. We add some comments on why all should be considered canonical.

Keywords: linear transforms, canonical transforms, Lie group Sp$(2,R)$.

DOI: https://doi.org/10.3842/SIGMA.2012.033

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Full text: http://emis.mi.ras.ru/.../033
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Bibliographic databases:

ArXiv: 1206.1123
MSC: 20C10; 20C35; 33C15; 33C45
Received: April 24, 2012; in final form June 1, 2012; Published online June 6, 2012
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Citation: Kurt Bernardo Wolf, “A top-down account of linear canonical transforms”, SIGMA, 8 (2012), 033, 13 pp.

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Mielnik B., “Quantum Operations: Technical Or Fundamental Challenge?”, J. Phys. A-Math. Theor., 46:38 (2013), 385301  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. B. Mielnik, “Quantum control: discovered, repeated and reformulated ideas mark the progress”, 8th International Symposium on Quantum Theory and Symmetries (QTS8), Journal of Physics Conference Series, 512, IOP Publishing Ltd, 2014, 012035  crossref  mathscinet  isi  scopus
    3. Arik S.O., Ozaktas H.M., “Optimal Representation and Processing of Optical Signals in Quadratic-Phase Systems”, Opt. Commun., 366 (2016), 17–21  crossref  adsnasa  isi  scopus
    4. K. B. Wolf, “Development of linear canonical transforms: A historical sketch”, Linear Canonical Transforms, Springer Series in Optical Sciences, 198, eds. J. Healy, M. Kutay, H. Ozaktas, J. Sheridan, Springer-Verlag Berlin, 2016, 3–28  crossref  mathscinet  zmath  isi  scopus
    5. R. Andriambololona, R. T. Ranaivoson, H. D. E. Randriamisy, H. Rakotoson, “Dispersion operators algebra and linear canonical transformations”, Int. J. Theor. Phys., 56:4 (2017), 1258–1273  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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