Szegő kernel

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^''n'' with ''C''² boundary, and let ''A''(Ω) denote the space of all holomorphic functions in Ω that are continuous on $\overline$. Define the Hardy space ''H''²(∂Ω) to be the closure in ''L''²(∂Ω) of the restrictions of elements of ''A''(Ω) to the boundary. The Poisson integral implies that each element ''ƒ'' of ''H''²(∂Ω) extends to a holomorphic function ''Pƒ'' in Ω. Furthermore, for each ''z'' ∈ Ω, the map
:$f\mapsto Pf(z)$
defines a continuous linear functional on ''H''²(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel ''k''_''z'', which is to say
:$Pf(z) = \int_ f(\zeta)\overline\,d\sigma(\zeta).$

The Szegő kernel is defined by
:$S(z,\zeta) = \overline,\quad z\in\Omega,\zeta\in\partial\Omega.$
Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in ''z''. In fact, if ''φ''_''i'' is an orthonormal basis of ''H''²(∂Ω) consisting entirely of the restrictions of functions in ''A''(Ω), then a Riesz–Fischer theorem argument shows that
:$S(z,\zeta) = \sum_^\infty \phi_i(z)\overline.$

References

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{{DEFAULTSORT:Szego kernel
Complex analysis
Several complex variables