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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| Видавець: | Інститут математики НАН України |
|---|---|
| Дата: | 2011 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2011
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147171 |
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| 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. |
Репозиторії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-147171 |
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dspace |
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irk-123456789-1471712019-02-14T01:26:14Z The Fourier U(2) Group and Separation of Discrete Variables Wolf, K.B. 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. 2011 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147171 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Wolf, K.B. |
| spellingShingle |
Wolf, K.B. The Fourier U(2) Group and Separation of Discrete Variables Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Wolf, K.B. |
| author_sort |
Wolf, K.B. |
| title |
The Fourier U(2) Group and Separation of Discrete Variables |
| title_short |
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_sort |
fourier u(2) group and separation of discrete variables |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147171 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:26:46Z |
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2023-05-20T17:26:46Z |
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