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A unified approach to atomic decompositions via integrable group representations. (English) Zbl 0658.22007

Function spaces and applications, Proc. US-Swed. Semin., Lund/Swed., Lect. Notes Math. 1302, 52-73 (1988).
[For the entire collection see Zbl 0633.00013.]
The basic idea underlying coherent state representations is to reconstruct a given function f from its coefficients \((<f| g_ j>)_{j\in I}\) with respect to a family of functions \((g_ j)_{j\in I}\) derived from a single reference function g by a linear group action. Examples are the Gabor representation which plays an important role in the field of neural networks for image processing [cf. J. G. Daugman, Relaxation neural networks for complete discrete 2-d Gabor transforms. Visual Communications and Image Processing ’88, SPIE 1001, 1048-1061 (1988)], and the wavelet representation with applications to seismic signal processing [cf. J. Morlet, Sampling theory and wave propagation. In: Issues in Acoustic Signal-Image Processing and Recognition, NATO ASI Series, F1, 233-261 (1983)]. The associated linear groups are the Heisenberg group, or more precisely, the polarized section to the center in the Heisenberg group [cf. the reviewer’s monograph entitled “Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory” (1986; Zbl 0632.43001)], and the solvable exponential affine Lie group of the real line \({\mathbb{R}}\), respectively [cf. G. Warner, Harmonic analysis on semi-simple Lie groups I (1972; Zbl 0265.22020)] with their unitary duals parametrized by the Kirillov coadjoint orbits. - In the paper under review, the authors describe a general functional analytic approach to coherent state representations which covers the examples mentioned above.
Reviewer: W.Schempp

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
81R30 Coherent states
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
22D10 Unitary representations of locally compact groups
43A80 Analysis on other specific Lie groups
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
46J10 Banach algebras of continuous functions, function algebras