Extensions of the Heisenberg group and wavelet analysis in the plane (Q2715888)

\noindent A third of the human cortex cerebri is devoted to vision, which is the most important sense of human beings. Radar extends the visional sense. The radar pioneer C.A. Wiley first observed that a side--looking radar system can improve its azimuth resolution by utilizing the Doppler spread of the echo signal [E.N. Leith, Synthetic aperture radar. In: Optical Data Processing, D. Casasent, editor, pp. 89--117, Topics in Applied Physics, Vol. 23, Springer--Verlag, Berlin, Heidelberg, New York 1978; E.N. Leith, Optical processing of synthetic aperture radar data. In: Photonic Aspects of Modern Radar, H. Zmuda, E.N. Toughlian, editors, pp. 381--401, Artech House, Boston, London 1994]. His landmark observation that the key to better vision is a larger radar antenna aperture signified the birth of a technology now referred to as synthetic aperture radar (SAR) image formation [C.A. Wiley, Synthetic aperture radars, IEEE Trans. Aerosp. Electron. Syst., Vol. AES--21, 440--443 (1985)]. The principal idea behind the SAR imaging modality to synthesize the effect of a large--aperture physical radar system whose physical radar antenna construction is infeasible has also been used since the former radar engineer Sir Martin Ryle (Nobel prize 1974) introduced the Very Large Array (VLA) technique into astrophysics. The synthetic aperture technique of VLA which is based on the Kepplerian moving platform method finally led to the Very Long Baseline Array (VLBA), and the Very Long Baseline Interferometry (VLBI) of quasars and radio galaxies in modern radio astronomy. Because the rotating planar coadjoint orbits of the three--dimensional simply connected two--step nilpotent Heisenberg Lie group \(N\) provide a mathematical model for VLA, VLBA, and VLBI by means of the spatial matched filter bank configuration of the unitary dual \(\hat{N}\) [R. Penrose, The apparent shape of a relativistically moving sphere. Proc. Cambridge Phil. Soc. 55, 137--139 (1959); W. Schempp, Harmonic Analysis on the Heisenberg Nilpotent Lie Group, with Applications to Signal Theory. Pitman Research Notes in Mathematics Series, Vol. 147, Longman Scientific and Technical, London (1986; Zbl 0632.43001); R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions. The International Series of Monographs on Chemistry, Vol. 14, Oxford University Press, Oxford, New York, Toronto 1991; A.W. Rihaczek, Principles of High--Resolution Radar. Artech House, Boston, London (1996; Zbl 0874.94003); W.J. Schempp, Magnetic Resonance Imaging: Mathematical Foundations and Applications. Wiley--Liss, New York, Chichester, Weinheim (1998; Zbl 0930.92015); A. Terras, Fourier Analysis on Finite Groups and Applications. Cambridge University Press, Cambridge, New York, Melbourne (1999; Zbl 0928.43001)], extensions of \(N\) have been studied within the context of wavelet theory and vision [J. Segman, W. Schempp, On the extension of the Heisenberg group to incorporate multiscale resolution. In: Wavelets and their Applications, J.S. Byrnes, J.L. Byrnes, K.A. Hargreaves, and K. Berry, editors, Kluwer Academic Publishers, NATO ASI Ser., Ser. C., Math. Phys. Sci. 442, 347--361 (1994; Zbl 0816.42020)]. The paper under review describes a class of closed subgroups \(\bigl\{\,G_p \,| \, p \in [-1,+1]\bigr\}\) of the general linear group \(GL(3,{\mathbf R})\) formed as semi--direct products with the strictly positive real numbers \({\mathbf R}^{\times}_+\) acting on \(N\) via dilations that are diagonal with respect to the polarized parametrization of \(N\). For parameter values \(p \in \, ]-1,+1]\), the square integrable representations of the four--dimensional simply connected solvable Lie groups \(G_p\) with extended Heisenberg Lie algebra spanned by the real matrices NEWLINE\[NEWLINE\begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix},\qquad\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix},\qquad\begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix},\qquad\begin{pmatrix} p + 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0\end{pmatrix}, NEWLINE\]NEWLINE then lead to continuous wavelet transforms acting on square integrable functions on the real plane \({\mathbf R} \oplus {\mathbf R}\).NEWLINENEWLINEFor the entire collection see [Zbl 0911.00013].