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

( B f ) ( z ) = ∫ D ( 1 − | z | 2 ) 2 | 1 − z w ¯ | 4 f ( w ) d A ( w ) , {\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 d A {\displaystyle \mathrm {d} A} is the area measure. It is named after Felix Alexandrovich Berezin.

References

  • 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.

External links

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


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