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Szegő kernel

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In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician Gábor Szegő.

Let Ω be a bounded domain in C with C boundary, and let A(Ω) denote the space of all holomorphic functions in Ω that are continuous on Ω ¯ {\displaystyle {\overline {\Omega }}} . Define the Hardy space H(∂Ω) to be the closure in L(∂Ω) of the restrictions of elements of A(Ω) to the boundary. The Poisson integral implies that each element ƒ of H(∂Ω) extends to a holomorphic function Pƒ in Ω. Furthermore, for each z ∈ Ω, the map

f ↦ P f ( z ) {\displaystyle f\mapsto Pf(z)}

defines a continuous linear functional on H(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel kz, which is to say

P f ( z ) = ∫ ∂ Ω f ( ζ ) k z ( ζ ) ¯ d σ ( ζ ) . {\displaystyle Pf(z)=\int _{\partial \Omega }f(\zeta ){\overline {k_{z}(\zeta )}}\,d\sigma (\zeta ).}

The Szegő kernel is defined by

S ( z , ζ ) = k z ( ζ ) ¯ , z ∈ Ω , ζ ∈ ∂ Ω . {\displaystyle S(z,\zeta )={\overline {k_{z}(\zeta )}},\quad z\in \Omega ,\zeta \in \partial \Omega .}

Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in z. In fact, if φi is an orthonormal basis of H(∂Ω) consisting entirely of the restrictions of functions in A(Ω), then a Riesz–Fischer theorem argument shows that

S ( z , ζ ) = ∑ i = 1 ∞ ϕ i ( z ) ϕ i ( ζ ) ¯ . {\displaystyle S(z,\zeta )=\sum _{i=1}^{\infty }\phi _{i}(z){\overline {\phi _{i}(\zeta )}}.}

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