复Minkowski空间中推广的MacMahon-Schwinger主定理与共形波
Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space
摘要：
在1+3维共形群SO（4，2）（局部同构于SU（2，2））的齐次空间（Cartan域）D4 =SO（4，2）/（SO（4）\times SO（2））上构造了连续小波变换（CWT）.流形D_4可以通过Cayley变换一一映射到复Minkowski空间的未来管域C^4_+上，在文献中已经提出了其他类型的（电磁）小波。本文研究了Hilbert空间L^2_h上共形群的酉不可约表示（D_4，d ν_λ）和L^2_h（C^4_+，d \tilde ν_λ）与连续质量谱的同构性，证明了（等方差）之间的希尔伯特空间，容许性和紧框架条件，给出了齐次多项式的重构公式和正交基，讨论了所提出的共形小波的对称性和欧氏极限。为此，我们首先陈述并证明了Schwinger主定理（SMT）的一个λ-推广，它是一个有用的数学工具，特别是作为共形群的酉表示函数的生成函数和L^2_h（D_4，d ν_λ）的再生（Bergman）核的导出。SMT与MacMahon的主定理（MMT）有关，为了完整性，还提供了Louck的SU（N）固体谐波方面的扩展。并研究了收敛条件。
英文摘要：
We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times 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^2_h(D_4,dν_λ) and L^2_h(C^4_+,d\tildeν_λ) of square integrable holomorphic functions with scale dimension λand 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 L^2_h(D_4,dν_λ). 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.
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