PlanetMath: Bargmann transform

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Bargmann transform

(Definition)

The Bargmann transform of a function, , is a linear map $B: X(\mathbb{R}) \to Y(\mathbb{C})$ defined by

$\displaystyle B f(z)=\sqrt{2} \int_\mathbb{R} f(t) \hskip2pt e^{2\pi t z-\pi t^2 - \frac{\pi}{2} z^2}\, dt$

Theorem 1 The Bargmann transform on $L^2(\mathbb{R})$ , $B: L^2(\mathbb{R}) \to \mathcal{F}^2(\mathbb{C})$ , is a unitary transformation. Here $\mathcal{F}^2(\mathbb{C})$ is the Fock space.

Bibliography

1: Karlheinz Gröchenig, "Foundations of Time-Frequency Analysis," Birkhhäuser (2000)

"Bargmann transform" is owned by ErlendA.

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Bargmann transform

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Cross-references: Fock space, unitary transformation, linear map, function

This is version 4 of Bargmann transform, born on 2007-02-24, modified 2007-02-24.
Object id is 8970, canonical name is BargmannTransform.
Accessed 720 times total.

Classification:

AMS MSC:

43A15 (Abstract harmonic analysis :: $L^p$-spaces and other function spaces on groups, semigroups, etc.)

Pending Errata and Addenda

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