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Zbl 0608.46014
Daubechies, Ingrid; Grossmann, A.; Meyer, Y.
Painless nonorthogonal expansions. (English)
[J] J. Math. Phys. 27, 1271-1283 (1986). ISSN 0022-2488; ISSN 1089-7658/e

In a Hilbert space ${\cal H}$, discrete families of vectors $\{h\sb j\}$ with the property that $f=\sum\sb{j}<h\sb j\vert f>h\sb j$ for every f in ${\cal H}$ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ${\cal H}$, but also can hold in situations where the $h\sb j$ 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 2000:: *46C99 Inner product spaces, Hilbert spaces
46B15 Summability and bases in normed spaces

Keywords: orthonormal basis; overcomplete; Weyl-Heisenberg coherent states; affine coherent states; quasiorthogonal expansions

Cited in: Zbl 1196.65003 Zbl 1113.42001

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