Extended MacMahon–Schwingerʼs Master Theorem and conformal wavelets in complex Minkowski space
We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D 4 = SO ( 4 , 2 ) / ( SO ( 4 ) × SO ( 2 ) ) of the conformal group SO ( 4 , 2 ) (locally isomorphic to SU ( 2 , 2 ) ) in 1 + 3 dimensions. The manifold D 4 can be mapped one-to-one onto the future tube doma...
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| Published in: | Applied and computational harmonic analysis Vol. 31; no. 1; pp. 143 - 168 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
01.07.2011
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| Subjects: | |
| ISSN: | 1063-5203, 1096-603X |
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| Abstract | We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain)
D
4
=
SO
(
4
,
2
)
/
(
SO
(
4
)
×
SO
(
2
)
)
of the conformal group
SO
(
4
,
2
)
(locally isomorphic to
SU
(
2
,
2
)
) in
1
+
3
dimensions. The manifold
D
4
can be mapped one-to-one onto the future tube domain
C
+
4
of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces
L
h
2
(
D
4
,
d
ν
λ
)
and
L
h
2
(
C
+
4
,
d
ν
˜
λ
)
of square integrable holomorphic functions with scale dimension
λ and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a
λ-extension of Schwingerʼs Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of
L
h
2
(
D
4
,
d
ν
λ
)
. SMT is related to MacMahonʼs Master Theorem (MMT) and an extension of both in terms of Louckʼs
SU
(
N
)
solid harmonics is also provided for completeness. Convergence conditions are also studied. |
|---|---|
| AbstractList | We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain)
D
4
=
SO
(
4
,
2
)
/
(
SO
(
4
)
×
SO
(
2
)
)
of the conformal group
SO
(
4
,
2
)
(locally isomorphic to
SU
(
2
,
2
)
) in
1
+
3
dimensions. The manifold
D
4
can be mapped one-to-one onto the future tube domain
C
+
4
of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces
L
h
2
(
D
4
,
d
ν
λ
)
and
L
h
2
(
C
+
4
,
d
ν
˜
λ
)
of square integrable holomorphic functions with scale dimension
λ and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a
λ-extension of Schwingerʼs Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of
L
h
2
(
D
4
,
d
ν
λ
)
. SMT is related to MacMahonʼs Master Theorem (MMT) and an extension of both in terms of Louckʼs
SU
(
N
)
solid harmonics is also provided for completeness. Convergence conditions are also studied. |
| Author | Pérez-Romero, E. Calixto, M. |
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| CitedBy_id | crossref_primary_10_1088_1751_8113_47_11_115302 crossref_primary_10_1016_j_geomphys_2019_103505 crossref_primary_10_1016_j_laa_2013_09_043 crossref_primary_10_1088_1751_8113_48_49_495304 crossref_primary_10_1007_JHEP05_2025_091 |
| Cites_doi | 10.1137/0401043 10.1006/aama.1996.0008 10.1063/1.531623 10.1112/plms/s2-8.1.223 10.1109/TFTSA.1992.274200 10.1142/S0129055X90000028 10.1088/0305-4470/27/15/022 10.1112/plms/s2-8.1.77 10.1016/j.acha.2005.05.001 10.1007/s00041-003-0009-x 10.1007/BF01645506 10.1063/1.532481 10.1063/1.526761 10.1006/acha.1994.1012 10.1006/aima.1998.1769 10.1016/j.acha.2009.10.002 10.1006/acha.1999.0272 10.1088/0305-4470/36/30/201 10.1007/s00041-010-9142-5 10.1063/1.1288497 10.1016/0003-4916(85)90306-9 10.1007/s00041-008-9027-z 10.1073/pnas.0606003103 10.1016/j.acha.2006.02.001 10.1088/0305-4470/27/20/017 10.1016/S1063-5203(02)00507-9 10.1016/j.acha.2008.04.004 10.1016/j.acha.2007.01.003 |
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| Keywords | Complex domains Non-commutative harmonic analysis Conformal group Schwingerʼs Master Theorem Unitary holomorphic representations Continuous wavelet transform |
| Language | English |
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| Snippet | We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain)
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| SubjectTerms | Complex domains Conformal group Continuous wavelet transform Non-commutative harmonic analysis Schwingerʼs Master Theorem Unitary holomorphic representations |
| Title | Extended MacMahon–Schwingerʼs Master Theorem and conformal wavelets in complex Minkowski space |
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