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Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups

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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.

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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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  1. Laboratory for Computational Sensing and Robotics (LCSR), Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, USA

    Arash Ghaani Farashahi

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  1. Arash Ghaani Farashahi
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Correspondence to Arash Ghaani Farashahi.

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Communicated by David Kimsey.

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Ghaani Farashahi, A. Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups. Complex Anal. Oper. Theory 12, 1537–1548 (2018). https://doi.org/10.1007/s11785-017-0717-x

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