Expanding a Finite Blaschke Product

The little Bloch space, B0, is the space of all holomorphic functions f on the unit disk such that limlzl?1 lf'(z)l (1- lzl2) = 0. Finite Blaschke products are clearly in B0, but examples of infinite products in B0 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others). Stephenson has asked whether B0 contains an infinite, indestructible Blaschke product, i.e., a Blaschke product B so that (B(z) - a)/(1 - âB(z)), is also a Blaschke product for every element a I D. In this paper we give an afirmative answer to his question by constructing such a Blaschke product. We also answer a question of Carmona and Cufi by constructing a VMO function, f, so that ll f ll8 = 1 and whose range set, R(f,a) = {w : there exists zn ? a, f(zn) = w}, equals the open unit disk for every a I T.

It is shown that rational functions continuous on the unit circle and of maximal equios-cillation order are either nonconstant finite Blaschke products or nonconstant inverted finite Blaschke products.

For any finite Blaschke product B, there is an injective analytic map $$\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}$$ and a polynomial p of the same degree as B such that $$B=p\circ \varphi $$ on $${\mathbb {D}}$$ . Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial p or the associated conformal map $$\varphi $$ . In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary finite degree whose zeros are equally spaced on a circle centered at the origin.

The new framework of random polynomials developed by R. Pemantle, I. Rivin and the late O. Schramm has been studied in this thesis. The strong Pemantle-Rivin conjecture asks whether for random polynomials with independent and identically distributed zeros with a common probability distribution μon the complex plane, the empirical measures of their critical points would converge weakly to μ almost surely. This convergence question has connection with geometry of polynomials. S. D. Subramanian confirmed the conjecture whenμis a non-uniform distribution supported in the unit circle ∂D.
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\nIn this thesis, the conjecture has been extended to considering not only the critical points of the random polynomials, but also the zeros of their higher order, polar and Sz.-Nagy's generalized derivatives. The case thatμ(uniform or not) is supported in ∂D has been studied, where the derivative of each order has been proved to satisfy the conjecture. Subramanian's work has thereby been completed plus generalization. The same almost sure weak convergence has also been shown for polar and Sz.-Nagy's generalized derivatives, under some mild conditions. In particular, the result on polar derivative is the crux of filling up Subramanian's missing case of uniform μ.
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\nMeanwhile, the original Pemantle{Rivin conjecture asks about convergence in probability instead of the aforesaid stronger almost sure convergence. Z. Kabluchko fully solved this conjecture. In this thesis, his methodologies have been adapted to prove the analogous conjecture for random finite Blaschke products.
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\nMore precisely, for random finite Blaschke products with independent and identically distributed zeros with a common probability distributionμon the unit disc, the empirical measures of their critical points have been shown to converge weakly toμin probability. Consequently, this work contributes the very first probabilistic result to T. W. Ng and C. Y. Tsang's polynomial-finite-Blaschke-product dictionary.
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\nWhile the above works address probabilistic problems about zero distribution of polynomials and finite Blaschke products (regarded as non-euclidean polynomials by J. L. Walsh), the other part of this thesis is an application of the BKK theory in deterministic polynomial systems.
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\nThe numbers of various configurations for vortex dynamics in an ideal plane fluid without background flow had been investigated by many researchers for decades. In this thesis, fixed equilibria have been studied in the presence of a steady, incompressible and irrotational background flow. A more physically significant definition of a fixed equilibrium configuration has been suggested. Under this new definition, an attainable generic upper bound for their number has been found (in terms of the degree of the non-constant polynomial that determines the background flow and of the sizes of vortex species).
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\nOur result is established by transforming the rational function system arisen from fixed equilibria into a polynomial system, whose form is good enough to apply the aforesaid BKK theory (named after D. N. Bernshtein, A. G. Khovanskii and A. G. Kushnirenko) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem.

Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

We study geometrical properties of finite Blaschke products. For a Blaschke product B of degree d, let \(L_{\lambda }\) be the set of the lines tangent to the unit circle at the d preimages \( B^{-1}(\lambda ) \). We show that the trace of the intersection points of each pair of two elements in \( L_{\lambda } \) as \( \lambda \) ranges over the unit circle forms an algebraic curve of degree at most \( d-1 \). In case of low degree, we have more precise results. For instance, for \( d=3 \), the trace forms a conic section. For \( d=4 \), we provide a necessary and sufficient condition for Blaschke products whose trace include a conic section.

An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space ${\mathcal {B}_0}$. In addition to showing how an inner function can meet the geometric conditions for ${\mathcal {B}_0}$, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.

Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^{n}$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where $z_{0}$ is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of $z_{0}$. In particular, we find how many curves there are that are fixed by $f$ and that land at $z_{0}$.

In this thesis, we characterize the Jacobson radical of the semicrossed product A(D)×α Z+ of the disk algebra by an endomorphism α where α is defined by the composition with a finite Blaschke product φ. Precisely, the Jacobson radical is the set of those whose 0th Fourier coefficient is identically zero and whose kth Fourier coefficient vanishes on the set of recurrent points of φ. Moreover, if φ is elliptic, i.e., φ has a fixed point in the open unit disc, then the Jacobson radical coincides with the set of quasinilpotent elements.

Uniform algebras that can be expressed as inductive limits of standard simpler algebras are of particular interest. For instance, some G-disc algebras are inductive limits of sequences of disc algebras, connected with finite Blaschke products, called also Blaschke inductive limit algebras. Here we show, among others, that only G-disc algebras, and the spaces \( H^\infty (\mathbb{D}_G ) \) with G ⊂ ℚ, can be expressed as limits of countable inductive sequences of algebras of type \( A(\mathbb{D}) \) and H∞ correspondingly. We study also inductive limits of sequences of spaces of type H∞ and prove corresponding corona theorems. Further, we establish relationships between Bourgain algebras of coordinate algebras, and the Bourgain algebra of their inductive limit, and also between H∞-spaces on \( \mathbb{D} \), as coordinate algebras in an inductive sequence, and their inductive limit. While we state all results for general shift-invariant algebras AS, they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras.

The classical Gauss—Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain “locally inner” self-maps of the disk. More precisely, we look at a unit-norm function f ∈ H∞ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, I, is nontrivial. Under certain conditions to be discussed, it follows that f′ must also have a nontrivial inner factor, say J, and we study the relationship between the boundary singularities of I and J. Examples are furnished to show that our sufficient conditions cannot be substantially relaxed.

A Solution of max |<i>a</i>₁| Using Finite Blaschke Products

Let F and G be finite Blaschke products of degree greater than one. Let p be the Wolff point of F such that either or . We prove that F commutes with G if and only if for some positive integers i,j. Also, if F and G have the same degree and commute, we prove that either F = G or G = ωF, where ω is a root of the unity.

In this paper we consider the following ‘Toeplitz completion’ problem: Complete the unspecified analytic Toeplitz entries of the partial block Toeplitz matrix $ \begin{linenomath} A:=\begin{bmatrix} T_{\overline\psi_1}&amp; ?\\[4pt] \T_{\overline\psi_2} \end{bmatrix} \end{linenomath} $ to make A hyponormal, where ψi∈H∞ is a non-constant rational function for i=1,2.