# 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 GrÃÂ¶chenig, ”Foundations of Time-Frequency Analysis,” BirkhhÃÂ¤user (2000)
Title Bargmann transform BargmannTransform 2013-03-22 16:44:45 2013-03-22 16:44:45 ErlendA (6587) ErlendA (6587) 7 ErlendA (6587) Definition msc 43A15 Bargmann transform