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 ...
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Problem in understanding Blaschke product.
Blaschke product is defined in the following way $:$
$$B(z) = \prod\limits_{n = 1}^{\infty} \frac {|z_n|} {z_n} \frac {z_n - z} {1 - \overline z_n z},\ z_n \in \mathbb D \setminus \{\textbf 0\}\ \...
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Expression for Finite Blaschke Products
We know that every proper analytic map from the unit disk to itself is a finite Blaschke product, i.e.
$$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}$$ with $a, b, c, d\in\mathbb{R}$ and $ad-bc=1.$
Also, on the unit disk, ...
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Blaschke product having poles proof
I am studying Banach Spaces of Analytic Function by Hoffman. Hoffman proves the following theorem:
The Blaschke product whose zeroes are \begin{align*} \alpha_{1} ,
\alpha_2 , \ldots \end{align*} ...
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If sequence of finite Blaschke Products $(B_n)_{n\in N}$ converges uniformly to $B$ then $\frac{B_n(z)}{z^m}$ converges uniformly to $\frac{B}{z^m}$
I am having problem with showing that if sequence of finite Blaschke Products
$$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}$$ I have used $\frac{d}{dz}\prod_{m=1}^nf_m(z)=(\prod_{m=1}^nf_m(z))(\sum_{m=1}^n\frac{f_m'(z)}{f_m(z)})=f(z)(\sum_{m=...
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Julia set of finite Blaschke product
I want to compute the Julia set of finite Blaschke product $B_3$ from the second version of the paper "PARABOLIC AND NEAR-PARABOLIC RENORMALIZATION FOR LOCAL DEGREE THREE" by FEI YANG,, ...
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The relationship between Blaschke products and the Poisson kernel.
I'm reading an approximation result from a paper that claims without justifying:
$$\cfrac{d}{d\theta}\text{arg}(B(e^{i\theta}))=\sum_{j=1}^nP(e^{i\theta},a_j),$$
where $z=re^{i\phi}$ and $P(z,a_j)=\...
user879426
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Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product
Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...
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Can a Blaschke product have unrestricted limit equal to zero
By a theorem in Hoffmans book we know that a Blaschke product $B(z)$ is analytic in the closed unit disc everywhere except the compact set $K$ which consists of the accumulation of it's zeros. However ...
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Derivative of Blaschke product
Let $z_n$ be a Blaschke sequence in $\mathbb{D}$ and let $B$ be the Blaschke product defined by $$B(z)=z^m\prod_{n=1}^{\infty}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\bar{z}_nz}$$
I'm trying to show the ...
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Application of Montel's Theorem
I am working on the following problem:
Let $\mathcal{F}$ denote the set of functions which are analytic on a neighborhood of the closed unit disk in $\mathbb{C}$. Find:
$$\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 = ...
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