The Fourier U(2) Group and Separation of Discrete Variables
The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,R), whose maximal compact subgroup is the Fourier group U(2)F; this includes isotropic and anisotropic Fourier transforms, screen rotations and gyrations in the phase space of ray positions and op...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2011 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2011
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147171 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Fourier U(2) Group and Separation of Discrete Variables / K.B. Wolf, L.E. Vicent // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862689003663261696 |
|---|---|
| author | Wolf, K.B. |
| author_facet | Wolf, K.B. |
| citation_txt | The Fourier U(2) Group and Separation of Discrete Variables / K.B. Wolf, L.E. Vicent // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,R), whose maximal compact subgroup is the Fourier group U(2)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 so(4). Two distinct subalgebra chains are used to model arrays of N² points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N2-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.
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| first_indexed | 2025-12-07T16:10:16Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147171 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:10:16Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wolf, K.B. 2019-02-13T18:08:04Z 2019-02-13T18:08:04Z 2011 The Fourier U(2) Group and Separation of Discrete Variables / K.B. Wolf, L.E. Vicent // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20F28; 22E46; 33E30; 42B99; 78A05; 94A15 DOI:10.3842/SIGMA.2011.053 https://nasplib.isofts.kiev.ua/handle/123456789/147171 The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,R), whose maximal compact subgroup is the Fourier group U(2)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 so(4). Two distinct subalgebra chains are used to model arrays of N² points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N2-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. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
 We thank the support of the Optica Matem´atica ´ projects DGAPA-UNAM IN-105008 and SEPCONACYT 79899, and we thank Guillermo Kr¨otzsch (ICF-UNAM) for his assistance with the graphics and Juvenal Rueda-Paz (Facultad de Ciencias, Universidad Aut´onoma del Estado de Morelos) for his support with the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Fourier U(2) Group and Separation of Discrete Variables Article published earlier |
| spellingShingle | The Fourier U(2) Group and Separation of Discrete Variables Wolf, K.B. |
| title | The Fourier U(2) Group and Separation of Discrete Variables |
| title_full | The Fourier U(2) Group and Separation of Discrete Variables |
| title_fullStr | The Fourier U(2) Group and Separation of Discrete Variables |
| title_full_unstemmed | The Fourier U(2) Group and Separation of Discrete Variables |
| title_short | The Fourier U(2) Group and Separation of Discrete Variables |
| title_sort | fourier u(2) group and separation of discrete variables |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147171 |
| work_keys_str_mv | AT wolfkb thefourieru2groupandseparationofdiscretevariables AT wolfkb fourieru2groupandseparationofdiscretevariables |