Interpolating Blaschke Products: Stolz and Tangential Approach Regions

Girela, Daniel; Peláez, José Ángel; Vukotić, Dragan

doi:10.1007/s00365-006-0651-6

Cited by 13 publications

(10 citation statements)

References 30 publications

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“…In particular, A p 0 is the classical Bergman space A p . The last assertion in Theorem The question of when the derivative of a Blaschke product with zeros in a Stolz angle belongs to the weighted Bergman spaces has been studied, for example, in [24][25][26].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Inner functions in the Möbius invariant Besov-type spaces

Pérez-González

Rättyä

2009

Proceedings of the Edinburgh Mathematical Society

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An analytic function f in the unit disc D belongs to F (p, q, s)is uniformly bounded for all a ∈ D. Here g(z, a) = − log |ϕa(z)| is the Green function of D, and ϕa(z) = (a − z)/(1 −āz). It is shown that for 0 < γ < ∞ and |w| = 1 the singular inner function exp(γ(z + w)/(z − w)) belongs to F (p, q, s), 0 < s 1, if and only if p q + 1 2 (s + 3). Moreover, it is proved that, if 0 < s < 1, then an inner function belongs to the Möbius invariant Besov-type space B p s = F (p, p − 2, s) for some (equivalently for all) p > max{s, 1 − s} if and only if it is a Blaschke product whose zero sequence {zn} satisfies sup a∈D ∞ n=1 (1 − |ϕa(zn)| 2 ) s < ∞.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Inner functions in the Möbius invariant Besov-type spaces

Pérez-González

Rättyä

2009

Proceedings of the Edinburgh Mathematical Society

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“…We now assume m > 0 and apply the envelope algorithm to find the discriminant envelope of the family of circles in (10). In fact, we prove: Proof.…”

Section: Proof Of Theorem 25mentioning

confidence: 96%

Applications of envelopes

Bickel

Gorkin

Tran

2020

Complex Anal Synerg

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Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. Here we use envelopes of families of circles to study objects from matrix theory and hyperbolic geometry. First we explore relationships between numerical ranges of 2 × 2 matrices and families of circles to study the elliptical range theorem. Then we deduce a relationship between envelopes and the boundaries of families of intersecting circles and use it to find the boundaries of various families of pseudohyperbolic disks.

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“…Note that in the case γ = 1 we have to assume C > 1. For γ > 1, R(γ, ξ, C) is a tangential approaching region in D, which touches T at ξ. Denote by R γ the family of all Blaschke products whose zeros lie in some R(γ, ξ, C) with a fixed γ. References related to R γ are for instance [4,11,21]. With these preparations we are ready to state our first main result.…”

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confidence: 89%

Remarks on one-component inner functions

Reijonen¹

2019

Ann. Acad. Sci. Fenn. Math.

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A one-component inner function Θ is an inner function whose level set Ω Θ (ε) = {z ∈ D : |Θ(z)| < ε} is connected for some ε ∈ (0, 1). We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for 0 < p < ∞, the derivative of a one-component inner function Θ is a member of the Hardy space H p if and only if Θ ′′ belongs to the Bergman space A p p−1 , or equivalently Θ ′ ∈ A 2p p−1. 1. Examples of one-component inner functions Let D be the open unit disc of the complex plane C. A bounded and analytic function in D is an inner function if it has unimodular radial limits almost everywhere on the boundary T of D. In this note, we study so-called one-component inner functions [14], which are inner functions Θ whose level set Ω Θ (ε) = {z ∈ D : |Θ(z)| < ε} is connected for some ε ∈ (0, 1). In particular, Blaschke products in this class are of interest. For a given sequence {z n } ⊂ D \ {0} satisfying n (1 − |z n |) < ∞, the Blaschke product with zeros {z n } is defined by B(z) = n |z n | z n z n − z 1 − z n z , z ∈ D. Here each zero z n is repeated according to its multiplicity. In addition, we assume that {z n } is ordered by non-decreasing moduli. Recently several authors have studied one-component inner functions in the context of model spaces and operator theory; see for instance [6, 8, 9, 10]. In addition, Aleksandrov's paper [5], which contains several characterizations for one-component inner functions, is worth mentioning. These references do not offer any concrete examples of infinite one-component Blaschke products; even though, reference [5] offers tools for this purpose. In recent paper [13] by Cima and Mortini, one can find some examples. However, all one-component Blaschke products constructed in [13] have some heavy restrictions. Roughly speaking, zeros of all of them are at least uniformly separated. Recall that {z n } ⊂ D is called uniformly separated if inf n∈N k =n z k − z n 1 − z k z n > 0.

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Interpolating Blaschke Products: Stolz and Tangential Approach Regions

Cited by 13 publications

References 30 publications

Inner functions in the Möbius invariant Besov-type spaces

Inner functions in the Möbius invariant Besov-type spaces

Applications of envelopes

Remarks on one-component inner functions

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