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SIGMA, 2011, Volume 7, 053, 18 pp. (Mi sigma611)  

This article is cited in 4 scientific papers (total in 4 papers)

The Fourier $\mathsf U(2)$ Group and Separation of Discrete Variables

Kurt Bernardo Wolfa, Luis Edgar Vicentb

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

Abstract: The linear canonical transformations of geometric optics on two-dimensional screens form the group $\mathsf{Sp}(4,\mathfrak R)$, whose maximal compact subgroup is the Fourier group $\mathsf U(2)_\mathrm F$; this includes isotropic and anisotropic Fourier transforms, screen rotations and gyrations in the phase space of ray positions and optical momenta. Deforming classical optics into a Hamiltonian system whose positions and momenta range over a finite set of values, leads us to the finite oscillator model, which is ruled by the Lie algebra $\mathsf{so}(4)$. Two distinct subalgebra chains are used to model arrays of $N^2$ points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The $N^2$-vectors in this space are digital (pixellated) images on either of these two grids, related by a unitary transformation. Here we examine the unitary action of the analogue Fourier group on such images, whose rotations are particularly visible.

Keywords: discrete coordinates; Fourier $\mathsf U(2)$ group; Cartesian pixellation; polar pixellation

DOI: https://doi.org/10.3842/SIGMA.2011.053

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Full text: http://emis.mi.ras.ru/journals/SIGMA/2011/053/
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Bibliographic databases:

ArXiv: 1106.0093
MSC: 20F28; 22E46; 33E30; 42B99; 78A05; 94A15
Received: February 19, 2011; in final form May 26, 2011; Published online June 1, 2011
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Citation: Kurt Bernardo Wolf, Luis Edgar Vicent, “The Fourier $\mathsf U(2)$ Group and Separation of Discrete Variables”, SIGMA, 7 (2011), 053, 18 pp.

Citation in format AMSBIB
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\by Kurt Bernardo Wolf, Luis Edgar Vicent
\paper The Fourier $\mathsf U(2)$ Group and Separation of Discrete Variables
\jour SIGMA
\yr 2011
\vol 7
\papernumber 053
\totalpages 18
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\crossref{https://doi.org/10.3842/SIGMA.2011.053}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2804583}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855218148}


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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. Gibbons G.W., Rugina C., “Goryachev-Chaplygin, Kovalevskaya, and Brdicka-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stackel-Killing tensors”, J Math Phys, 52:12 (2011), 122901  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Bernardo Wolf K., “Group Theory in Finite Hamiltonian Systems”, Symmetries in Science XV, Journal of Physics Conference Series, 380, IOP Publishing Ltd, 2012, 012004  crossref  mathscinet  isi  scopus
    3. Bernardo Wolf K., Atakishiyev N.M., “The Fourier U(2)(F) Group on Square and Round Pixellated Arrays”, 7th International Conference on Quantum Theory and Symmetries (Qts7), Journal of Physics Conference Series, 343, IOP Publishing Ltd, 2012, 012130  crossref  isi  scopus
    4. Bernardo Wolf K., “Royal Road From Geometric To Discrete Optics”, Photonics Lett. Pol., 7:1 (2015), 5–7  crossref  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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