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 Bƒ : D → C defined at a point z ∈ D by

${\displaystyle (Bf)(z)=\int _{D}{\frac {(1-|z|^{2})^{2}}{|1-z{\bar {w}}|^{4}}}f(w)\,\mathrm {d} A(w),}$

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

• Hedenmalm, Haakan; Korenblum, Boris; Zhu, Kehe (2000). Theory of Bergman spaces. Graduate Texts in Mathematics. Vol. 199. New York: Springer-Verlag. pp. 28–51. ISBN  0-387-98791-6. MR  1758653.

• Weisstein, Eric W. "Berezin transform". MathWorld.

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