Bergman kernel

In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cⁿ.

In detail, let L²(D) be the Hilbert space of square integrable functions on D, and let L^2,h(D) denote the subspace consisting of holomorphic functions in D: that is,

where H(D) is the space of holomorphic functions in D. Then L^2,h(D) is a Hilbert space: it is a closed linear subspace of L²(D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D

${\displaystyle \sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{2}(D)}}$

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