1. See also

2. References

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

for every compact subset K of D. Thus convergence of a sequence of holomorphic functions in L²(D) implies also compact convergence, and so the limit function is also holomorphic.

Another consequence of ({{EquationRef|1}}) is that, for each z ∈ D, the evaluation

is a continuous linear functional on L^2,h(D). By the Riesz representation theorem, this functional can be represented as the inner product with an element of L^2,h(D), which is to say that

The Bergman kernel K is defined by

The kernel K(z,ζ) holomorphic in z and antiholomorphic in ζ, and satisfies

One key observation about this picture is that L^2,h(D) may be identified with the space of holomorphic (n,0)-forms on D, via multiplication by . Since the inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated Bergman metric are therefore automatically invariant under the automorphism group of the domain.

See also

• Bergman metric
• Bergman space
• Szegő kernel

References

• {{Citation | last1=Krantz | first1=Steven G. | authorlink=Steven Krantz|title=Function Theory of Several Complex Variables | publisher=American Mathematical Society | location=Providence, R.I. | isbn=978-0-8218-2724-6 | year=2002}}.
• {{springer|title=Bergman kernel function|id=B/b015560|first=E.M.|last=Chirka}}.

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