##
**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.

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 |