Ambiguity functions, Wigner distributions and Cohen's class for LCA groups
type: Item
instance of: scholarly article
title: Ambiguity functions, Wigner distributions and Cohen's class for LCA groups
zbMATH Open document ID: 1032.43002
DOI: 10.1016/S0022-247X(02)00630-3
author: Gitta Kutyniok
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-04-28T00:00:00Z
review text: The simply connected three-dimensional Heisenberg two-step nilpotent Lie group \(N = ({\mathbb R} \oplus {\mathbb R})\vartriangleleft{\mathbb R}\) implements in a natural way the automorphism group of dilations \((\delta_\lambda)_{\lambda > 0}\) which act via multiplication with \(\lambda\) on the two coordinates of the plane \({\mathbb R} \oplus {\mathbb R}\), and via multiplication with \(\lambda^2\) on the central line \({\mathbb R}\) of \(N\). The irreducible unitary linear representations of \(N\) which admit a dimension \(> 1\) are square integrable modulo the one-dimensional center [cf. \textit{C. C. Moore} and \textit{J. A. Wolf}, Trans. Am. Math. Soc. 185, 445-462 (1973; Zbl 0274.22016)]. The framework of an abelian group \(G\) suggests the substitution of the dilation property of \(N\) by the assumption that the mapping \(x \mapsto x^2\) forms an automorphism of \(G\). Under the weaker assumption of 2-root-compactness, the paper under review develops a commutative theory of ambiguity functions and Wigner distributions for a class of locally compact abelian topological groups \(G\) with dual group \(\widehat{G}\) which parallels to a certain extent the two-step nilpotent case [\textit{E. P. Wigner}, Quantum-mechanical distribution functions revisited, in: Perspectives in Quantum Theory, pp. 25-36 (1979); \textit{W. Schempp}, Proc. Am. Math. Soc. 92, 103-110 (1984; Zbl 0525.43007)]. For applications of time-frequency analysis, however, the two-step nilpotent case remains the most important one. The reader should consult [\textit{W. Schempp}, Harmonic Analysis on the Heisenberg Nilpotent Lie Group, with Applications to Signal Theory (1986); \textit{S. Thangavelu}, Lectures on Hermite and Laguerre Expansions (1993); \textit{W. Schempp}, Wavelet modelling of clinical magnetic resonance tomography: An ensemble quantum computing approach. In: Inverse Problems, Tomography, and Image Processing, pp. 129-176 (1998); \textit{W. J. Schempp}, Magnetic Resonance Imaging…
Mathematics Subject Classification ID: 43A25
zbMATH DE Number: 1901856
reviewed by: Walter Schempp
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901856