Mathematical theorem
In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981) and A. I. Markushevich (1908–1979) in 1934, is a result concerning the approximation in mean square of holomorphic functions on a bounded open set in the complex plane by complex polynomials. It states that complex polynomials form a dense subspace of the Bergman space of a domain bounded by a simple closed Jordan curve. The Gram–Schmidt process can be used to construct an orthonormal basis in the Bergman space and hence an explicit form of the Bergman kernel, which in turn yields an explicit Riemann mapping function for the domain.
Proof.
Let Ω be the bounded Jordan domain and let Ω"n" be bounded Jordan domains decreasing to Ω, with Ω"n" containing the closure of Ω"n" + 1. By the Riemann mapping theorem there is a conformal mapping "f""n" of Ω"n" onto Ω, normalised to fix a given point in Ω with positive derivative there. By the Carathéodory kernel theorem "f""n"("z") converges uniformly on compacta in Ω to "z". In fact Carathéodory's theorem implies that the inverse maps tend uniformly on compacta to "z". Given a subsequence of "f""n", it has a subsequence, convergent on compacta in Ω. Since the inverse functions converge to "z", it follows that the subsequence converges to "z" on compacta. Hence "f""n" converges to "z" on compacta in Ω.
As a consequence the derivative of "f""n" tends to 1 uniformly on compacta.
Let "g" be a square integrable holomorphic function on Ω, i.e. an element of the Bergman space A2(Ω). Define "g""n" on Ω"n" by "g""n"("z") = "g"("f""n"("z"))"f""n"'("z"). By change of variable
formula_1
Let "h""n" be the restriction of "g""n" to Ω. Then the norm of "h""n" is less than that of "g""n". Thus these norms are uniformly bounded. Passing to a subsequence if necessary, it can therefore be assumed that "h""n" has a weak limit in A2(Ω). On the other hand, "h""n" tends uniformly on compacta
to "g". Since the evaluation maps are continuous linear functions on A2(Ω), "g" is the weak limit of "h""n". On the other hand, by Runge's theorem, "h""n" lies in the closed subspace "K" of "A"2(Ω) generated by complex polynomials. Hence "g" lies in the weak closure of "K", which is "K" itself.
Notes.
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