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2013 | Buch

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Blaschke Products and Their Applications

herausgegeben von: Javad Mashreghi, Emmanuel Fricain

Verlag: Springer US

Buchreihe : Fields Institute Communications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Blaschke Products and Their Applications presents a collection of survey articles that examine Blaschke products and several of its applications to fields such as approximation theory, differential equations, dynamical systems, harmonic analysis, to name a few. Additionally, this volume illustrates the historical roots of Blaschke products and highlights key research on this topic. For nearly a century, Blaschke products have been researched. Their boundary behaviour, the asymptomatic growth of various integral means and their derivatives, their applications within several branches of mathematics, and their membership in different function spaces and their dynamics, are a few examples of where Blaschke products have shown to be important. The contributions written by experts from various fields of mathematical research will engage graduate students and researches alike, bringing the reader to the forefront of research in the topic. The readers will also discover the various open problems, enabling them to better pursue their own research.

Inhaltsverzeichnis

Frontmatter

Applications of Blaschke Products to the Spectral Theory of Toeplitz Operators

Abstract

The chapter is a survey of some applications of Blaschke products to the spectral theory of Toeplitz operators. Topics discussed include Toeplitz operators with bounded measurable symbols, factorisation with an infinite index, compositions with Blaschke products, representation of functions with a given asymptotic behaviour of the argument in a neighbourhood of a discontinuity in the form of a composition of a continuous function with a Blaschke product, and applications to the KdV equation.

Sergei Grudsky, Eugene Shargorodsky

Approximating the Riemann Zeta-Function by Strongly Recurrent Functions

Abstract

Bhaskar Bagchi has shown that the Riemann hypothesis holds if and only if the Riemann zeta-function ζ(z) is strongly recurrent in the strip 1/2<ℜz<1. In this note we show that ζ(z) can be approximated by strongly recurrent functions sharing important properties with ζ(z).

P. M. Gauthier

A Survey on Blaschke-Oscillatory Differential Equations, with Updates

Abstract

In the celebrated 1949 paper due to Nehari, necessary and sufficient conditions are given for a locally univalent meromorphic function to be univalent in the unit disc $\mathbb{D}$. The proof involves a second order differential equation of the form

$$ f''+A(z)f=0, $$

(†)

where A(z) is analytic in $\mathbb{D}$. As an immediate consequence of the proof, it follows that if |A(z)|≤1/(1−|z|²)² for every $z\in\mathbb{D}$, then any non-trivial solution of (†) has at most one zero in $\mathbb{D}$.

Since 1949 a number of papers provide with different types of growth conditions on the coefficient A(z) such that the solutions of (†) have at most finitely many zeros in $\mathbb{D}$. If there exists at least one solution with infinitely many zeros in $\mathbb{D}$, then (†) is oscillatory. If the zeros still satisfy the classical Blaschke condition, then (†) is called Blaschke-oscillatory. This concept was introduced by the author in 2005, but the topic was considered by Hartman and Wintner already in 1955 (Trans. Am. Math. Soc. 78:492–500). This semi-survey paper provides with a collection of results and tools dealing with Blaschke-oscillatory equations.

As for results, necessary and sufficient conditions are given, and notable effort has been put in dealing with prescribed zero sequences satisfying the Blaschke condition. The concept of Blaschke-oscillation also extends to differential equations of arbitrary order. Many of the results given in this paper have been published earlier in a weaker form. All questions regarding the zeros of solutions can be rephrased for the critical points of solutions. This gives rise to a new concept called Blaschke-critical equations. To intrigue the reader, several open problems are pointed out in the text.

Some classical tools and closely related topics that are often related to the finite oscillation case include the Schwarzian derivative, properties of univalent functions, Green’s identity, conformal mappings, and a certain Hardy-Littlewood inequality. The Blaschke-oscillatory case also makes use of interpolation theory, various growth estimates for logarithmic derivatives of Blaschke products, Bank-Laine functions and recently updated Wiman-Valiron theory.

Janne Heittokangas

Bi-orthogonal Expansions in the Space L 2(0,∞)

Abstract

In this paper we deduce bi-orthogonal expansions in the space L ²(0,∞) with respect to two special systems of functions from the corresponding expansions in the Hardy space $H^{2}_{+}$ for the upper half-plane.

André Boivin, Changzhong Zhu

Blaschke Products as Solutions of a Functional Equation

Abstract

In this paper, a family of Blaschke products, as non-trivial inner solutions of Schröder’s equation, is introduced. This observation leads to the construction of a surjective composition operator on an infinite dimensional model subspace of H ².

Javad Mashreghi

Cauchy Transforms and Univalent Functions

Abstract

We use a formula of Pommerenke relating the primitives of functions which are the Cauchy transforms of measures on the unit circle to their behavior in the space of functions of bounded mean oscillation. This is a linear process and it has some smoothness. Further, there is a non-linear map from the Cauchy transforms into the normalized univalent functions. We show that for the subspace H ¹ of Cauchy transforms the univalent functions so obtained have quasi-conformal extensions to all of the plane.

Joseph A. Cima, John A. Pfaltzgraff

Critical Points, the Gauss Curvature Equation and Blaschke Products

Abstract

In this survey paper we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger–Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique “maximal” Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties similar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.

Daniela Kraus, Oliver Roth

Growth, Zero Distribution and Factorization of Analytic Functions of Moderate Growth in the Unit Disc

Abstract

We give a survey of results on zero distribution and factorization of analytic functions in the unit disc in classes defined by the growth of log|f(re ^iθ)| in the uniform and integral metrics. We restrict ourself to the case of finite order of growth. For a Blaschke product B we obtain a necessary and sufficient condition for the uniform boundedness of all p-means of log|B(re ^iθ)|, where p>1.

Igor Chyzhykov, Severyn Skaskiv

Hardy Means of a Finite Blaschke Product and Its Derivative

Abstract

In this chapter we consider several topics related to finite Blaschke products $B_{n}(z)=\prod_{k=1}^{n}\frac{z_{k}-z}{1-\overline{z}_{k} z}$ in the unit disc of the complex plane and their Hardy means $M_{p}^{p}(r,B)=\frac{1}{2\pi}\int^{2\pi}_{0}|B(re^{i\theta})|^{p} d\theta$. We discuss two explicit formulae for $1-M_{2}^{2}(r,B)$: when B has distinct zeroes or a single zero repeated n times. We relate the growth of the means $M_{2}^{2}(r,B)$ and $M_{2}^{2}(r,B^{\prime})$ to “sampling means” $\sum^{n}_{k=1}|B(rz_{k})|(1-|z_{k}|^{2})$ and $\sum^{n}_{k=1}|B^{\prime}(rz_{k})|(1-|z_{k}|^{2})$. It is shown, for products of degree two and three, that if the zeroes lie on the circle of radius |z|=ρ<1 with constant angle ϕ between successive zeroes, then $1-M_{2}^{2}(r,B)$ is an increasing function of ϕ. We conjecture that this holds true for products of arbitrary finite degree.

Alan Gluchoff, Frederick Hartmann

Hyperbolic Derivatives Determine a Function Uniquely

Abstract

The notion of hyperbolic derivative for functions from the unit disc to itself is well known. Recently, Rivard has proposed a definition for higher-order derivatives. We prove that the sequence of hyperbolic derivatives of order n (n=0,1,2,…) of a function f determines this function uniquely.

Line Baribeau

Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane

Abstract

A multiresolution analysis in the Hardy space of the unit disc was introduced recently (see Pap in J. Fourier Anal. Appl. 17(5):755–776, 2011). In this paper we will introduce an analogous construction in the Hardy space of the upper half plane. The levels of the multiresolution are generated by localized Cauchy kernels on a special hyperbolic lattice in the upper half plane. This multiresolution has the following new aspects: the lattice which generates the multiresolution is connected to the Blaschke group, the Cayley transform and the hyperbolic metric. The second: the nth level of the multiresolution has finite dimension (in classical affine multiresolution this is not the case) and still we have the density property, i.e. the closure in norm of the reunion of the multiresolution levels is equal to the Hardy space of the upper half plane. The projection operator to the nth resolution level is a rational interpolation operator on a finite subset of the lattice points. If we can measure the values of the function on the points of the lattice the discrete wavelet coefficients can be computed exactly. This makes our multiresolution approximation very useful from the point of view of the computational aspects.

Hans G. Feichtinger, Margit Pap

Norms of Composition Operators Induced by Finite Blaschke Products on Möbius Invariant Spaces

Abstract

We obtain an asymptotic formula for the norms of composition operators induced by finite Blaschke products on analytic (quotient) Besov spaces in terms of their degree. We also compute the norms of such operators on the true Bloch and Dirichlet spaces.

María J. Martín, Dragan Vukotić

On the Computable Theory of Bounded Analytic Functions

Abstract

The theory of bounded analytic functions is reexamined from the viewpoint of computability theory.

Timothy H. McNicholl

Polynomials Versus Finite Blaschke Products

Abstract

The aim of this chapter is to compare polynomials of one complex variable and finite Blaschke products and demonstrate that they share many similar properties. In fact, we collect many known results as well as some very recent results for finite Blaschke products here to establish a dictionary between polynomials and finite Blaschke products.

Tuen Wai Ng, Chiu Yin Tsang

Recent Progress on Truncated Toeplitz Operators

Abstract

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason’s seminal paper (Oper. Matrices 1(4):491–526, 2007).

Stephan Ramon Garcia, William T. Ross

Titel: Blaschke Products and Their Applications
herausgegeben von: Javad Mashreghi
Emmanuel Fricain
Verlag: Springer US
Electronic ISBN: 978-1-4614-5341-3
Print ISBN: 978-1-4614-5340-6
DOI: https://doi.org/10.1007/978-1-4614-5341-3