Ambiguity functions, Wigner distributions and Cohen's class for LCA groups
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: Mathematical Foundations and Applications (1998); \textit{E. Binz} and \textit{W. Schempp}, Quantum teleportation and spin echo: A unitary symplectic spinor approach. In: Aspects of Complex Analysis, Differential Geometry, Mathematical Physics and Applications, pp. 314-365 (1999); \textit{E. Binz} and \textit{W. Schempp}, Quantum hologram and relativistic hodogram: Magnetic resonance tomography and gravitational wavelet detection, in: Geometry, Integrability and Quantization, pp. 110-150 (2001); \textit{V. C. Chen} and \textit{L. Hao}, Time-Frequency Transforms for Radar Imaging and Signal Analysis (2002)]. Innovative optical applications in the two-step nilpotent framework are concerned with the orbital angular momentum of photons in laser light modes [\textit{L. Allen}, \textit{M. W. Beijersbergen}, \textit{R. J. C. Spreeuw} and \textit{J. P. Woerdman}, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A45, 8185-8189 (1992); \textit{N. B. Simpson}, \textit{K. Dholakia}, \textit{L. Allen} and \textit{M. J. Padgett}, Mechanical equivalence of spin and orbital angular momentum of light: an optical scanner, Opt. Lett. 22, 52-54 (1997); \textit{L. Allen}, \textit{M. J. Padgett} and \textit{M. Babiker}, The orbital angular momentum of light, in: Progress in Optics, Vol. XXXIX, 291-372 (1999); \textit{J. Leach}, \textit{M. J. Padgett}, \textit{S. M. Barnett} and \textit{S. Franke-Arnold}, J. Courtial, Measuring the orbital angular momentum of a single photon, Phys. Rev. Lett. 88, 257901-1 to 257901-4 (2002); \textit{S. Barreiro} and \textit{J. W. R. Tabosa}, Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms, Phys. Rev. Lett. 90, 133001-1 to 133001-4 (2003)].
scientific article; zbMATH DE number 1901856
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