Théorème de Farrell-Markushevich

Théorème de Farrell-Markushevich En mathématiques, le théorème de Farrell-Markushevich, prouvé indépendamment par O. J. Farrell (1899–1981)[1] et un. je. Markushevich (1908–1979) dans 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.

Contenu 1 Preuve 2 Voir également 3 Remarques 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, c'est à dire. an element of the Bergman space A2(Oh). Define gn on Ωn by gn(z) =g(fn(z))fn'(z). By change of variable {style d'affichage style d'affichage {|g_{n}|_{Oméga _{n}}^{2}=|g|_{Oméga }^{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(Oh). D'autre part, hn tends uniformly on compacta to g. Since the evaluation maps are continuous linear functions on A2(Oh), g is the weak limit of hn. D'autre part, by Runge's theorem, hn lies in the closed subspace K of A2(Oh) 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", Taureau. Amer. Math. Soc., 40: 908–914, est ce que je:10.1090/s0002-9904-1934-06002-6 Markushevich, UN. je. (1967), Theory of functions of a complex variable. Volume. III, Prentice–Hall Conway, John B. (2000), A course in operator theory, Études supérieures en mathématiques, volume. 21, Société mathématique américaine, ISBN 0-8218-2065-6 cacher vte Analyse fonctionnelle (sujets – glossaire) Espaces BanachBesovFréchetHilbertHölderNucléaireOrliczSchwartzSobolevvecteur topologique Propriétés tonneaucomplètedouble (algébrique/topologique)localement convexe réflexif séparable Théorèmes Hahn–Banach Représentation de Riesz graphe fermé principe de délimitation uniforme Kakutani virgule fixeKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Opérateurs adjointlimitécompactHilbert–Schmidtnormalnucléairetraceclasstransposéillimitéunitaire problème de sous-espaceconjecture de MahlerApplicationsespace de Hardythéorie spectrale des équations différentielles ordinairesnoyau de chaleurthéorème d'indexcalcul des variationscalcul fonctionnelopérateur intégralpolynôme de Jonesthéorie des champs quantiques topologiquesgéométrie non commutativehypothèse de Riemanndistribution (ou fonctions généralisées) Sujets avancés propriété d'approximationensemble équilibréThéorie de Choquettopologie faibleDistance de Banach–MazurThéorie de Tomita–Takesaki Catégories: Theorems in functional analysisOperator theoryTheorems in complex analysis