ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS
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Niveau: Supérieur, Doctorat, Bac+8
ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. We study the convergence of a sequence of finite Blaschke products of a fix order toward a rotation. This would enable us to get a better picture of a characterization theorem for finite Blaschke products. 1. Introduction Let (zk)1≤k≤n be a finite sequence in the open unit disc D. Then the rational function B(z) = ? n∏ k=1 zk ? z 1? z¯k z , where ? is a unimodular constant, is called a finite Blaschke product of order n for D [8]. There are various results characterizing these functions. For example, one of the oldest ones is due to Fatou. Theorem A (Fatou [5]). Let f be analytic in the open unit disc D and suppose that lim |z|?1 |f(z)| = 1. Then f is a finite Blaschke product. For an analytic function f : ?1 ?? ?2, the number of solutions of the equation f(z) = w, (z ? ?1, w ? ?2), counting multiplicities, is called the valence of f at w and is denoted by vf (w). It is well-known that a finite Blaschke product of order n has the constant valence n for each w ? D.
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ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. We study the convergence of a sequence of finite Blaschke products of a fix order toward a rotation. This would enable us to get a better picture of a characterization theorem for finite Blaschke products. 1. Introduction Let (zk)1≤k≤n be a finite sequence in the open unit disc D. Then the rational function B(z) = ? n∏ k=1 zk ? z 1? z¯k z , where ? is a unimodular constant, is called a finite Blaschke product of order n for D [8]. There are various results characterizing these functions. For example, one of the oldest ones is due to Fatou. Theorem A (Fatou [5]). Let f be analytic in the open unit disc D and suppose that lim |z|?1 |f(z)| = 1. Then f is a finite Blaschke product. For an analytic function f : ?1 ?? ?2, the number of solutions of the equation f(z) = w, (z ? ?1, w ? ?2), counting multiplicities, is called the valence of f at w and is denoted by vf (w). It is well-known that a finite Blaschke product of order n has the constant valence n for each w ? D.
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ON A CHARACTERIZATION OF FINITE BLASCHKE PRODUCTS
EMMANUEL FRICAIN, JAVAD MASHREGHI
Abstract.We study the convergence of a sequence of finite Blaschke products of a fix order toward a rotation. This would enable us to get a better picture of a characterization theorem for finite Blaschke products.
1.Introduction Let (zk)1≤k≤nbe a finite sequence in the open unit discDthe rational. Then function n Y zk−z B(z) =γ , 1−z¯kz k=1 whereγis a unimodular constant, is called a finite Blaschke product of ordernfor D[8]. There For example, one ofare various results characterizing these functions. the oldest ones is due to Fatou.
Theorem A(Fatou [5]).Letfbe analytic in the open unit discDand suppose that lim|f(z)|= 1. |z|→1 Thenfis a finite Blaschke product.
For an analytic functionf: Ω1−→Ω2, the number of solutions of the equation f(z) =w,(z∈Ω1, w∈Ω2), counting multiplicities, is called thevalenceoffatwand is denoted byvf(w). It is well-known that a finite Blaschke product of ordernhas the constant valencen for eachw∈D. But, this property in fact characterizes finite Blaschke products of ordern.
2000Mathematics Subject Classification.Secondary: 32A70.Primary: 30D50, Key words and phrases.Blaschke products, zero sets, convergence, automorphism. This work was supported by NSERC (Canada), Jacques Cartier Center and ANR project FRAB (France). 1
