Extended MacMahon-Schwinger's Master Theorem and conformal wavelets in complex Minkowski space

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D4=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 D4 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 Lh2(D4,dνλ) and Lh2(C+4,dν∼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 Schwinger's 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 Lh2(D4,d νλ). SMT is related to MacMahon's Master Theorem (MMT) and an extension of both in terms of Louck's SU(N) solid harmonics is also provided for completeness. Convergence conditions are also studied. © 2010 Elsevier Inc. All rights reserved.