Berezin transform explained

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

(B f)(z) = \int _{D} (( 1- |z| ^{2)} ^{2})/(| 1- z \bar{w) | ^{4}} f(w) d A (w),

where denotes the complex conjugate of w and

d A is the area measure. It is named after Felix Alexandrovich Berezin.

References

• Book: Hedenmalm , Haakan . Hakan Hedenmalm

. Hakan Hedenmalm. Korenblum, Boris . Boris Korenblum. Zhu, Kehe. Theory of Bergman spaces. Graduate Texts in Mathematics . 199. Springer-Verlag. New York. 2000. 28 - 51. 0-387-98791-6. 1758653.