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Discrete sets of coherent states and their use in signal analysis. (English) Zbl 0648.41017
Differential equations and mathematical physics, Proc. Int. Conf., Birmingham/Ala. 1986, Lect. Notes Math. 1285, 73-82 (1987).
[For the entire collection see Zbl 0619.00011.]
We discuss expansions of \(L^ 2\)-functions into \(\{\phi_{mn};m,n\in Z\}\), where the \(\phi_{mn}\) are generated from one function \(\phi\), either by translations in phase space, i.e. \(\phi_{mn}(x)=e^{imp_ ox}\phi (x-nq_ 0),\) \((p_ 0,q_ 0\) fixed), or by translations and dilations, i.e. \(\phi_{mn}(x)=a_ 0^{-m/2}\phi (a_ 0^{-m}x-nb_ 0).\) These expansions can be used for phase space localization.
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
phase space