Maximal Blaschke products

Kraus, Daniela; Roth, Oliver

doi:10.1016/j.aim.2013.03.017

Cited by 14 publications

(17 citation statements)

References 33 publications

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“…In [8], Kraus and Roth showed that a maximal Blaschke product F C extends analytically past any arc on the unit circle which does not meet the closure of C. We generalize their result to canonical solutions:…”

Section: When Are Canonical Solutions Maximal?mentioning

confidence: 78%

Critical structures of inner functions

Ivrii¹

2020

Preprint

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A celebrated theorem of M. Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. K. Dyakonov suggested that it may interesting to extend this result to infinite degree, however, one needs to be careful since inner functions may have identical critical sets. In this work, we try parametrizing inner functions by 1-generated invariant subspaces of the weighted Bergman space A 2 1 . Our technique is based on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.

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Section: When Are Canonical Solutions Maximal?mentioning

confidence: 78%

Critical structures of inner functions

Ivrii¹

2020

Preprint

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“…A result intimately connected to Theorem 1.1 has recently been proved in [6], see also [5]. There, so-called maximal Blaschke products have been studied.…”

Section: Remark 14 (Maximal Blaschke Products)mentioning

confidence: 94%

Composition and Decomposition of Indestructible Blaschke Products

Kraus

Roth

2013

Comput. Methods Funct. Theory

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“…This paper is expository, so there are essentially no proofs. For the proofs we refer to [39,40,41,42,44]. Background material on Hardy spaces and Bergman spaces can be found e.g.…”

Section: Critical Points Of Bounded Analytic Functionsmentioning

confidence: 99%

“…Theorem 3.6 (Semigroup property, [44]). The set of maximal Blaschke products is closed under composition.…”

Section: Related Extremal Problems Inmentioning

confidence: 99%

Critical Points, the Gauss Curvature Equation and Blaschke Products

Kraus

Roth

2013

Fields Institute Communications

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Abstract. In this survey paper we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger-Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique "maximal" Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties simliar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.

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Maximal Blaschke products

Cited by 14 publications

References 33 publications

Critical structures of inner functions

Critical structures of inner functions

Composition and Decomposition of Indestructible Blaschke Products

Critical Points, the Gauss Curvature Equation and Blaschke Products

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