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Painless nonorthogonal expansions. (English) Zbl 0608.46014
In a Hilbert space $ℋ$, discrete families of vectors $\left\{{h}_{j}\right\}$ with the property that $f={\sum }_{j}<{h}_{j}|f>{h}_{j}$ for every f in $ℋ$ are considered. This expansion formula is obviously true if the family is an orthonormal basis of $ℋ$, 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, Hilbert spaces 46B15 Summability and bases in normed spaces