Bargmann transform

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

$Bf(z)=\sqrt{2}\int_{\mathbb{R}}f(t)\hskip 2.0pte^{2\pi tz-\pi t^{2}-\frac{\pi}% {2}z^{2}}\,dt$

Theorem.

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.

References

1 Karlheinz GrÃÂ¶chenig, ”Foundations of Time-Frequency Analysis,” BirkhhÃÂ¤user (2000)

Title	Bargmann transform
Canonical name	BargmannTransform
Date of creation	2013-03-22 16:44:45
Last modified on	2013-03-22 16:44:45
Owner	ErlendA (6587)
Last modified by	ErlendA (6587)
Numerical id	7
Author	ErlendA (6587)
Entry type	Definition
Classification	msc 43A15
Defines	Bargmann transform