Abstract
Let G be a locally compact group, which need not be unimodular. Let x→U(x) (x∈G) be an irreducible unitary representation of G in a Hilbert space ℋ(U). Assume that U is square integrable, i.e., that there exists in ℋ(U) at least one nonzero vector g such that ∫‖(U(x)g,g)‖2 dx<∞. We give here a reasonably self-contained analysis of the correspondence associating to every vector f∈ℋ(U) the function (U(x)g,f) on G, discussing its isometry, characterization of the range, inversion, and simplest interpolation properties. This correspondence underlies many properties of generalized coherent states.
Keywords
Square-integrable functionMathematicsUnimodular matrixHilbert spaceUnitary representationIsometry groupGroup (periodic table)Isometry (Riemannian geometry)Group representationIrreducible representationPure mathematicsSquare (algebra)Unitary groupCombinatoricsUnitary stateLie groupPhysicsQuantum mechanics
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Publication Info
- Year
- 1985
- Type
- article
- Volume
- 26
- Issue
- 10
- Pages
- 2473-2479
- Citations
- 396
- Access
- Closed
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Cite This
A. Großmann,
J. Morlet,
T. Paul
(1985).
Transforms associated to square integrable group representations. I. General results.
Journal of Mathematical Physics
, 26
(10)
, 2473-2479.
https://doi.org/10.1063/1.526761
Identifiers
- DOI
- 10.1063/1.526761