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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.
Received: February 19, 2011; in final form May 26, 2011; Published online June 1, 2011
Citation:
Kurt Bernardo Wolf, Luis Edgar Vicent, “The Fourier $\mathsf U(2)$ Group and Separation of Discrete Variables”, SIGMA, 7 (2011), 053, 18 pp.
Linking options:
https://www.mathnet.ru/eng/sigma611 https://www.mathnet.ru/eng/sigma/v7/p53
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