scholarly journals Seeing the Monodromy Group of a Blaschke Product

2020 ◽  
Vol 67 (07) ◽  
pp. 1
Author(s):  
Elias Wegert
Keyword(s):  
Blaschke Product ◽  
Monodromy Group

A class of iterative greedy algorithms related to Blaschke product

2021 ◽  
Author(s):  
Tao Qian ◽  
Lihui Tan ◽  
Jiecheng Chen
Keyword(s):  
Blaschke Product ◽  
Greedy Algorithms

On a conjecture of Magnus on the Hurwitz Monodromy group

1973 ◽  
Vol 132 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Colin Maclachlan
Keyword(s):  
Monodromy Group

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

2021 ◽  
Vol 9 (1) ◽  
pp. 164-170
Author(s):  
Y. Gal ◽  
M. Zabolotskyi ◽  
M. Mostova
Keyword(s):  
Blaschke Product ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Counting Function ◽  
Finite System ◽  
Blaschke Products ◽  
Nevanlinna Characteristic ◽  
Number Of Zeros ◽  
Accumulation Points ◽  
Theory Of Value

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

1962 ◽  
Vol 14 ◽  
pp. 334-348 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Blaschke Product ◽  
Common Knowledge ◽  
Holomorphic Functions ◽  
Blaschke Products ◽  
Radial Limit ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

scholarly journals Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class C0

2015 ◽  
Vol 67 (1) ◽  
pp. 132-151
Author(s):  
Raphaël Clouâtre
Keyword(s):  
Blaschke Product ◽  
Main Tool ◽  
Unitary Equivalence ◽  
Finite Multiplicity ◽  
Minimal Function ◽  
Boundary Representations ◽  
Class C

AbstractWe obtain results on the unitary equivalence of weak contractions of class C0 to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool in this paper is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

The effect of rational substitutions on the monodromy group of the generaln th order equation of the Fuchsian class

1978 ◽  
Vol 163 (2) ◽  
pp. 111-119 ◽  
Author(s):  
Kathryn Kuiken
Keyword(s):  
Monodromy Group ◽  
Order Equation

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

Intersection of Two Invariant Subspaces

1987 ◽  
Vol 30 (2) ◽  
pp. 129-133 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Blaschke Product ◽  
Invariant Subspaces ◽  
Inner Functions

AbstractIt is shown that, if F and G are inner functions, (H2 ⊖ FH2)/(H2 ⊖ FH2) ∩ GH2 is n-dimensional if and only if G is a Blaschke product of degree n. This is an extension of the well known result for the case (H2 ⊖ FH2) ∩ GH2 = {0}.

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