Extensions of the Heisenberg group by dilations and frames
Theory and application of wavelet analysis have recently become a very active field of research because they are standing at the intersection of the frontiers of computerized Fourier analysis, algorithmic computing, computer controlled non-algorithmic parallel computing, massively parallel photonic signal processing, digital signal analysis, filter bank design, remote sensing, and visualization. From the mathematical point of view, the aforementioned disciplines overlap significantly. Over the years, however, the notational conventions and methodologies of the disciplines unfortunately diverged to the point that engineers and scientists working in one of the areas such as data compression, feature extraction, image processing, pattern recognition, flow visualization, synthetic aperture radar (SAR) imagery, voice signal processing, magnetic resonance imaging (MRI), or filter bank design often find similar or even identical work in the other of these areas nearly unrecognizable. Harmonic analysis on the Heisenberg nilpotent Lie group suggests to regard square integrability of continuous irreducible unitary group representations as an essential part of the Stone-von Neumann theorem of quantum mechanics. Then the nonlocality phenomenon of quantum physics is a consequence of quantum holography [\textit{W. Schempp}, Geometric analysis: The double-slit interference experimental and magnetic resonance imaging. In: Cybernetics and Systems `96, Vol. I, R. Trappl, Editor, pp. 179-183, Austrian Society for Cybernetic Studies, Vienna 1996]. The symplectic structure of the coadjoint orbits allows for implementation of the Keplerian temporospatial strategy used in physical astronomy for the synchronous and stroboscopic recording of phase histories in frequency subbands, and the design of multichannel coherent wavelet perfect reconstruction analysis-synthesis filter banks of matched filter type [\textit{W. Schempp}, The structure-function problem of Fourier transform magnetic resonance imaging. Wiley, New York (in print)]. The paper under review presents extensions of the Heisenberg group with dilations, and the affine group with modulations. Its focus is on two groups that contain affine and Heisenberg groups. The first one is the semidirect product of the Heisenberg group by dilations. Another way of extending the Heisenberg group by dilations is via the group of upper triangular matrices [\textit{J. Segman} and \textit{W. Schempp}, Two ways to incorporate scale in the Heisenberg group with an intertwining operator. J. Math. imaging Vis. 3, 79-94 (1993); also in: Wavelet Theory and Application, A. Laine, Editor, pp. 79-94 (1993; Zbl 0827.42018); On the extension of the Heisenberg group to incorporate multiscale resolution. In: Wavelets and their Applications, \textit{J. S. Byrnes} et al. (eds.) NATO ASI Ser., Ser. C, Math. Phys. Sci. 442, 347-361 (1994; Zbl 0816.42020)]. In the paper under review, generalized notions of square integrable group representations allow to fashion frames for \(L^2\) Hilbert spaces and \(L^p\) function spaces. Such frames combine advantages of the short-time Fourier transform and wavelet transform, and can be tailored to analyze specific types of signals.
scientific article; zbMATH DE number 772200