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Representations of SO(3) and angular polyspectra. (English) Zbl 1216.60027
Summary: We characterize the angular polyspectra, of arbitrary order, associated with isotropic fields defined on the sphere \(S^{2}=\{(x,y,z): x^{2}+ y^{2}+ z^{2}=1\}\). Our techniques rely heavily on group representation theory, and specifically on the properties of Wigner matrices and Clebsch-Gordan coefficients. The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the cosmic microwave background radiation, which is currently a crucial topic of investigation in cosmology. We also outline an application to random data compression and “simulation” of Clebsch-Gordan coefficients.

MSC:
60G10 Stationary stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
20C12 Integral representations of infinite groups
20C35 Applications of group representations to physics and other areas of science
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