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Painless nonorthogonal expansions. (English) Zbl 0608.46014
In a Hilbert space \({\mathcal H}\), discrete families of vectors \(\{h_ j\}\) with the property that \(f=\sum_{j}<h_ j| f>h_ j\) for every f in \({\mathcal H}\) are considered. This expansion formula is obviously true if the family is an orthonormal basis of \({\mathcal H}\), but also can hold in situations where the \(h_ j\) are not mutually orthogonal and are ”overcomplete”. The two classes of examples studied here are (i) appropriate sets of Weyl-Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ”quasiorthogonal expansions” will be a useful tool in many areas of theoretical physics and applied mathematics.

MSC:
46C99 Inner product spaces and their generalizations, Hilbert spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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