# Berezin transform

In mathematics — specifically, in complex analysis — the Berezin transform is an integral operator acting on functions defined on the open unit disk D of the complex plane C. Formally, for a function ƒ : D → C, the Berezin transform of ƒ is a new function  : D → C defined at a point z ∈ D by

$\displaystyle{ (B f)(z) = \int_D \frac{(1 - |z|^2)^2}{| 1 - z \bar{w} |^4} f(w) \, \mathrm{d}A (w), }$

where w denotes the complex conjugate of w and $\displaystyle{ \mathrm{d}A }$ is the area measure. It is named after Felix Alexandrovich Berezin.

## References

• Hedenmalm (2000). Theory of Bergman spaces. Graduate Texts in Mathematics. 199. New York: Springer-Verlag. pp. 28–51. ISBN 0-387-98791-6.