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The Slepian property for a class of time-frequency localization operators on the hyperbolic space. (English) Zbl 0888.43003

Let \(E\) denote the filtered Fourier transform. The problem of signal analysis of constructing the spectral resolution of the compact time-frequency localization operators \(E^*E\) and \(EE^*\) has been solved for the one-dimensional Fourier transform and a sharp edged filter by finding a second order linear differential operator that commutes with the integral operator \(E^*E\) and therefore admits the same eigenfunctions [D. Slepian, H. O. Pollak, Bell System Tech. J. 40, 43-63 (1961; Zbl 0184.08601); H. J. Landau, H. O. Pollak, Bell System Tech. J. 40, 65-84 (1961; Zbl 0184.08602); H. J. Landau, H. O. Pollak, Bell System Tech. J. 41, 1295-1336 (1962; Zbl 0184.08603)]. The purpose of the paper under review is an extension to more general classes of filters and radial functions on the three-dimensional hyperbolic space. The proof depends upon an isometric isomorphism transforming the space of radial square integrable functions on the unit ball \(\mathbb{B}_3\) under its Riemannian structure onto the analog space of functions on the three-dimensional Euclidean vector space.
Reviewer: W.Schempp (Siegen)

MSC:

43A32 Other transforms and operators of Fourier type
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
43A85 Harmonic analysis on homogeneous spaces
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