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Complex Analysis and Operator Theory

, Volume 12, Issue 7, pp 1537–1548 | Cite as

Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups

  • Arash Ghaani Farashahi
Article
  • 370 Downloads

Abstract

This paper presents theoretical aspects of a unified generalization for the abstract theory of coherent state/voice transforms over homogeneous spaces of compact groups using operator theory. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and \(\mu \) be the normalized G-invariant measure on G/H associated to the Weil’s formula with respect to the probability measures of G, H. Let \((\pi ,\mathcal {H}_\pi )\) be a continuous unitary representation of G with non-zero mean over H. In this article, we introduce the generalized notion of coherent state/voice transform associated to \(\pi \) on the Hilbert function \(L^2(G/H,\mu )\). We then study basic analytic properties of these transforms.

Keywords

Homogeneous space G-invariant measure Compact group Unitary representation Irreducible representation Coherent state/voice transform Inversion formula Resolution of the identity Reproducing kernel Hilbert spaces 

Mathematics Subject Classification

Primary 43A85 47A67 

Notes

Acknowledgements

The author would like to express his deepest gratitude to Prof. Hans G. Feichtinger for his valuable comments. Thanks are also due to Prof. S.T. Ali and also Prof. Hartmut Führ for stimulating discussions and pointing out various references during ICTP-TWAS school at ICTP.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratory for Computational Sensing and Robotics (LCSR), Whiting School of EngineeringJohns Hopkins UniversityBaltimoreUSA

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