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.


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


[1] DOI: 10.1090/S0002-9947-1952-0047179-6
[2] DOI: 10.1137/0515056 · Zbl 0578.42007
[3] DOI: 10.1137/0515056 · Zbl 0578.42007
[4] DOI: 10.1063/1.526761 · Zbl 0571.22021
[5] DOI: 10.1063/1.526761 · Zbl 0571.22021
[6] DOI: 10.1063/1.526761 · Zbl 0571.22021
[7] DOI: 10.1016/0034-4877(71)90006-1
[8] Perelomov A. M., Teor. Mat. Fiz. 6 pp 213– (1971)
[9] DOI: 10.1103/PhysRevB.12.1118
[10] DOI: 10.1103/PhysRevB.12.1118
[11] DOI: 10.1063/1.1664833 · Zbl 0184.54601
[12] DOI: 10.1063/1.526072
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