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# Bargmann transform

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

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

###### 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¶chenig, ”Foundations of Time-Frequency Analysis,” BirkhhÃ¤user (2000)

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Bargmann transform

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## Mathematics Subject Classification

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