Questions tagged [blaschke-products]

Use this tag for questions related to Blaschke products, which are bounded analytic functions in the open unit disk constructed to have zeros at a (finite or infinite) sequence of prescribed complex numbers inside the unit disk.

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Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints

It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
Math101's user avatar
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Problem in understanding Blaschke product.

Blaschke product is defined in the following way $:$ $$B(z) = \prod\limits_{n = 1}^{\infty} \frac {|z_n|} {z_n} \frac {z_n - z} {1 - \overline z_n z},\ z_n \in \mathbb D \setminus \{\textbf 0\}\ \...
Akiro Kurosawa's user avatar
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Expression for Finite Blaschke Products

We know that every proper analytic map from the unit disk to itself is a finite Blaschke product, i.e. $$f(z) = e^{i\theta}\prod_{i=1}^n \dfrac{z - a_i}{1 - \bar{a_i}z}$$ for some $\theta$ and some $...
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Orthogonal complement of $BH^2$, where B is Blaschke

It is known that if $B=\prod_{i=1}^{\infty} \frac{z-z_i}{1-\overline{z_i}z}$ where $z_1, z_2,...$ are all distinct and $\sum_{i=1}^{\infty}(1-|z_n|)<\infty$, then $(BH^2)^{\perp}=\overline{span}\{\...
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Analogue of Blaschke products for the upper half plane?

It is well-known that the biholomorphic self-maps of the upper half-plane are Mobius transformations $$\dfrac{az+b}{cz+d}$$ with $a, b, c, d\in\mathbb{R}$ and $ad-bc=1.$ Also, on the unit disk, ...
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Blaschke product: What does extending continuously mean here?

I am studying Factorisation of $H^p$ functions from Hoffman's Banach Spaces of Analytic Functions. The author is talking about extending the Blaschke product continuously to the accumulation points of ...
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Blaschke product having poles proof

I am studying Banach Spaces of Analytic Function by Hoffman. Hoffman proves the following theorem: The Blaschke product whose zeroes are \begin{align*} \alpha_{1} , \alpha_2 , \ldots \end{align*} ...
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If sequence of finite Blaschke Products $(B_n)_{n\in N}$ converges uniformly to $B$ then $\frac{B_n(z)}{z^m}$ converges uniformly to $\frac{B}{z^m}$

I am having problem with showing that if sequence of finite Blaschke Products $$B_n=z^m\prod_{k=1}^{n}\frac{|z_k|}{z_k}\frac{z_k - z}{1 - \overline{z}_k z}$$ where $(z_n)_{n\in \mathbb{N}} \subset \...
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Derivative of Blaschke product is not zero

To calculate the derivative of $$f(z)=\prod_{m=1}^n\frac{z-a_m}{1-\overline{a_m}z}$$ I have used $\frac{d}{dz}\prod_{m=1}^nf_m(z)=(\prod_{m=1}^nf_m(z))(\sum_{m=1}^n\frac{f_m'(z)}{f_m(z)})=f(z)(\sum_{m=...
stack_math's user avatar
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Julia set of finite Blaschke product

I want to compute the Julia set of finite Blaschke product $B_3$ from the second version of the paper "PARABOLIC AND NEAR-PARABOLIC RENORMALIZATION FOR LOCAL DEGREE THREE" by FEI YANG,, ...
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The relationship between Blaschke products and the Poisson kernel.

I'm reading an approximation result from a paper that claims without justifying: $$\cfrac{d}{d\theta}\text{arg}(B(e^{i\theta}))=\sum_{j=1}^nP(e^{i\theta},a_j),$$ where $z=re^{i\phi}$ and $P(z,a_j)=\...
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Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product

Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...
Ridwane El Mellass's user avatar
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Can a Blaschke product have unrestricted limit equal to zero

By a theorem in Hoffmans book we know that a Blaschke product $B(z)$ is analytic in the closed unit disc everywhere except the compact set $K$ which consists of the accumulation of it's zeros. However ...
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Derivative of Blaschke product

Let $z_n$ be a Blaschke sequence in $\mathbb{D}$ and let $B$ be the Blaschke product defined by $$B(z)=z^m\prod_{n=1}^{\infty}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\bar{z}_nz}$$ I'm trying to show the ...
Gregoire Rocheteau's user avatar
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Application of Montel's Theorem

I am working on the following problem: Let $\mathcal{F}$ denote the set of functions which are analytic on a neighborhood of the closed unit disk in $\mathbb{C}$. Find: $$\sup\{|f(0)|\mid f(1/2)=0=...
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If $\lim_{r \to 1} \frac{1}{2\pi}\int_0^{2\pi} \left| \log \left| f(re^{it}) \right| \right| dt = 0$ then $\left| f(z) \right| \leq 1$

My aim is to prove that Blaschke products are the only holomorphic funcions that verify the property $$ \lim_{r \to 1} \frac{1}{2\pi} \int_0^{2\pi} \left| \log \left| f(re^{it}) \right| \right| dt = ...
Javier Linares's user avatar
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How to Prove Uniform Convergence?

I was trying to prove If f is holomorphic on the open unit disk and|f(z)|= 1 for|z|= 1, then f is a finite Blaschke product. In the proof, I saw Since|f(z)|→1 uniformly as|z|→1, there is an r <1 so ...
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Find positive integers $n$ such that the derivative of $f(z)=(z-1)^nB(z)$ is bounded in $U$, where $B$ is a Blaschke product with zeros on $(0,1)$.

Suppose that $B$ is a Blaschke product with zeros on $(0,1)$. Find all $n \in \mathbb{N}$ such that the derivative of $f(z)=(z-1)^nB(z)$ is bounded in $U$. My thoughts: My guess is that it is true ...
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If $f$ is a finite Blaschke product then $f'(z) = 0$ if and only if $f'(\frac{1}{\bar{z}}) = 0$

How can I prove "if $f$ is a finite Blaschke product then $f'(z) = 0$ if and only if $f'(\frac{1}{\bar{z}}) = 0$"? A finite Blaschke product $f$ is a function of following type: $$f(z) = \alpha z^m \...
Vahid Shams's user avatar
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Real-valued bounded analytic functions on the unit disc

Let $f: \overline{\mathbb{D}} \to \mathbb{R}^+$ be a real (positive) valued function on the closed unit disc that is bounded and analytic on $\mathbb{D}$ (open unit disc) and $$\lim_{|z| \to 1}f(z) = ...
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Hankel operator with symbol a Blaschke product

If $B={\prod}_j \varphi_j$ is a Blaschke product (finite or infinite) of Blaschke factors $\varphi_j(w)=\frac{w-\alpha_j}{1-\overline{\alpha_j}w}$ with $|\alpha_j|>1$, is it true that the norm of ...
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Each point on the unit circle accumulation point of interpolating sequence?

I have an interpolating sequence for $H^\infty(\mathbb D)$ in the unit disk (that is, a uniformly separated sequence) which, in addition, satisfies the Blaschke condition. Is it possible that each ...
Friedrich Philipp's user avatar
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$\prod_{n=1}^\infty\frac{|a_n|}{a_n}\frac{z-a_n}{\overline{a}_n z-1}$ convergence

If $a_n\in\mathbb{C}$ are complex number such that $|a_n|<1$ and $\sum_{n}(1-|a_n|)<\infty$, then I know that following Blaschke product define an analytic function on the open unit disk $\...
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Zeros of bounded analytic function on unit disk -- infinite Blaschke product [duplicate]

Suppose we have $f: \mathbb{D} \rightarrow \mathbb{D}$ analytic and not identically zero. In order to prove $f$ has an infinite Blaschke product representation (where of course the product defines an ...
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Univalent Blaschke Products on $\mathbb{D}$

Let $B$ be a finite Blaschke product of degree $n$. Suppose there exist an open subset $U$ of $\mathbb{D}$ such that $B$ is univalent on $U.$ Is it true that $n=1$ ?
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Analytic function on unit disk has finitely many zeros

I am studying complex analysis from Theodore Gamelin's text and Exercise 1 of chapter IX.2 says that if $f$ is analytic inside the open unit disk and continuous on its boundary that satisfies $|f(z)| =...
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How can we draw a Blaschke $3$ ellipse?

Today I read the article Ellipses and Finite Blaschke Products by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini. In there they have proved very nice geometric results about per-images of Blaschke ...
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question on finite Blaschke product and its Julia set [closed]

I would like to make my previous question more precise. If $B$ is a finite Blaschke product such that its Julia set $J_B$ is a Cantor subset of $S^1$, then is it true that $B$ is expanding on $J_B\,$?...
Arkady Kitover's user avatar
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Limit of Blaschke product on the boundary of the unit disk

The following question was posed in Greene and Krantz's textbook 3rd edition, page 296 Problem 5: Let $\{a_n \}$ be a sequence in $\mathbb{D}$, the open unit disk s.t. $\sum_{n \in \mathbb{N}} 1 - |...
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Why (finite) Blaschke products are actually rational fractions?

I have found in several books the following affirmation : Let $f: \Delta \rightarrow \Delta$ be a non constant holomorphic function that extends continuously to $\overline{\Delta}$, $\Delta$ being ...
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