Farrell–Markushevich theorem

Farrell–Markushevich theorem In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981)[1] 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.

Contents 1 Proof 2 See also 3 Notes 4 References Proof Let Ω be the bounded Jordan domain and let Ωn be bounded Jordan domains decreasing to Ω, with Ωn containing the closure of Ωn + 1. By the Riemann mapping theorem there is a conformal mapping fn of Ωn onto Ω, normalised to fix a given point in Ω with positive derivative there. By the Carathéodory kernel theorem fn(z) converges uniformly on compacta in Ω to z.[2] In fact Carathéodory's theorem implies that the inverse maps tend uniformly on compacta to z. Given a subsequence of fn, 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 fn converges to z on compacta in Ω.

As a consequence the derivative of fn 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 gn on Ωn by gn(z) = g(fn(z))fn'(z). By change of variable {displaystyle displaystyle {|g_{n}|_{Omega _{n}}^{2}=|g|_{Omega }^{2}.}} Let hn be the restriction of gn to Ω. Then the norm of hn is less than that of gn. Thus these norms are uniformly bounded. Passing to a subsequence if necessary, it can therefore be assumed that hn has a weak limit in A2(Ω). On the other hand, hn tends uniformly on compacta to g. Since the evaluation maps are continuous linear functions on A2(Ω), g is the weak limit of hn. On the other hand, by Runge's theorem, hn 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.[3] See also Mergelyan's theorem Notes ^ Orin J. Farrell received his PhD (under J. L. Walsh) from Harvard University in 1930 and spent his career from 1931 at Union College with a leave of absence from January 1949 to May 1949 at the Institute for Advanced Study. See Orin J. Farrell at the Mathematics Genealogy Project; Bick, Theodore A. (1993). "A History of the Mathematics Department". Union College.; "Orin J. Farrell". Institute for Advanced Study. ^ See: Conway 2000, pp. 150–151 Markushevich 1967, pp. 31–35 ^ 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, vol. 21, American Mathematical Society, ISBN 0-8218-2065-6 hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Theorems in functional analysisOperator theoryTheorems in complex analysis