Farrell–Markushevich 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.[1] 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

${\displaystyle \displaystyle {\|g_{n}\|_{\Omega _{n}}^{2}=\|g\|_{\Omega }^{2}.}}$

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 A²(Ω) generated by complex polynomials. Hence g lies in the weak closure of K, which is K itself.[2]

See also

• Mergelyan's theorem

Notes

1. See:
   • Conway 2000, pp. 150–151
   • Markushevich 1967, pp. 31–35
2. Conway 2000, pp. 151–152

References

• Farrell, O. J. (1934), "On approximation to an analytic function by polynomials", Bull. Amer. Math. Soc., 40: 908–914, doi:10.1090/s0002-9904-1934-06002-6
• Markushevich, A. I. (1967), Theory of functions of a complex variable. Vol. III, Prentice–Hall
• Conway, John B. (2000), A course in operator theory, Graduate Studies in Mathematics, 21, American Mathematical Society, ISBN 0-8218-2065-6