IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
STOCKHOLM SWEDEN 2018,
Balschke Products and Their
Critical Values
JAKOB SUNDBLAD
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
IN
DEGREE PROJECT TEKNIK, FIRST CYCLE, 15 CREDITS STOCKHOLM SWEDEN 2018,
Blaschkeprodukter och
deras kritiska värden
JAKOB SUNDBLAD
KTH ROYAL INSTITUTE OF TECHNOLOGY SKOLAN FÖR TEKNIKVETENSKAP
Abstract
Abstrakt
Den här rapporten handlar om problemet att bestämma och beräkna en
Blaschkeprodukt från ett förutbestämt set av kritiska värden. Först presenteras en matematisk och teoretisk bakgrund till Balschkeprodukter, såsom Blaschkevillkoret, viktiga egenskaper hos ändliga Blaschke produkter och unikhetsteoremet. Problemet presenteras sedan och rapporten går igenom a beräkningsmetod som använder gaussisk kurvatur och teorin därbakom. Därefter, i samma kapitel, jämförs problemet med några ekvivalenta frågor. Avslutningsvis så presenteras två diskreta metoder för att lösa problemet. Den första använder teorin bakom cirkelpackning såsom det gjorde av Ken Stephenson och den andra använder numeriska metoder för att skriva om problemet till en kvadratisk matris av linjära ekvationer.
This paper concerns the problem of defining and computing a Blaschke product from a prescribed set of critical values. First a mathematical and theoretical background to Blaschke products and their critical values is presented, e.g. the Blaschke condition, important characteristics of finite Blaschke products and the uniqueness theorem. The problem is then presented and the paper walks through a computational method using gaussian curvature. In the same chapter the problem is then compared to some equivalent questions. Lastly two discrete methods are presented and compared. The first method uses the theory of circle packing as presented by Ken Stephenson and the second uses numerical methods to rewrite the problem as a quadratic matrix of linear equations.
2
Table of Contents
Introduction ... 3
1. Blaschke Products – a Definition ... 4
1.1 The Blaschke condition ... 4
1.2 Finite Blaschke products ... 4
1.3 Critical points ... 6
2. The Problem and its Analytical Solution ... 9
2.1 Gaussian curvature ... 9
2.2 Equivalence to other problems ... 11
2.2.1 Equilibrium position of point charge ... 11
3. Discrete Solutions ... 13
3.1 Circle packing ... 13
3.2 A quadratic model ... 15
References ... 17
Articles ... 17
Websites ... 17
3
Introduction
This paper is a literary study and aims to analyse and describe the problem of constructing a finite Blaschke product by studying its critical values. This will be done by first defining a finite Blaschke product in the complex plane and the mathematical background surrounding it. Using these definitions, the paper will formulate and describe the currently unsolved problem of constructing a finite Blaschke product by its critical values. Although there does not seem to exist an analytical solution to the problem the paper will in the last chapter present a few discrete methods using circle packing as presented in Ken Stephenson’s book Introduction to circle packing: the theory of discrete analytic function.
To understand the theory behind the problem some historical background may be useful.
The concept of Blaschke products were first introduced in 1915 by the Austrian
mathematician Wilhelm Blaschke in his paper Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen where he defined the Blaschke product as a bounded analytic function in the open unit disc with zeros at points (𝑎𝑛)
𝐵(𝑧) = 𝑒𝑖𝑎𝑧𝑘∏ |𝑎𝑛|
𝑎𝑛 𝑎𝑛−𝑧 1−𝑎̅̅̅̅𝑧𝑛
𝑛 (0.1)
Where 𝑘 ∈ ℕ0, 𝑎 ∈ ℝ and (𝑎𝑛) must fulfil the Blaschke condition
∑ (1 − |𝑎𝑛 𝑛|) < ∞ (0.2) 1
Because of the simple nature of the definition and the fact that it is derived from the zeros of the function Blaschke’s work lead to a new way of viewing other interesting holomorphic functions on the unit disc. A holomorphic function with a given set of zeros that fulfil the Blaschke condition (0.2) can be written as a product of a Blaschke product and another zero- free holomorphic function on the unit disc. In 1966 M.M. Dzhrbashyan found a more
generalized version of this. He constructed infinite products of a generalized version of Blaschke products with a condition that more easily converges. This allowed him to describe lager classes of meromorphic function in terms of these generalized Blaschke products. The property, that bear a lot of similarity to that of polynomials, has given Blaschke products a wide range of applications from approximation processes to optimization theory, the similarities to polynomials keep piling on, which brings us to the problem of this paper. Like for polynomials the ability to define a Blaschke function from its critical values would be a very useful tool and it would be significant to many parts of complex analysis.
1 W. Blaschke, "Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen" Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig , 67 (1915)
4
1. Blaschke Products – a Definition
1.1 The Blaschke condition
To understand Blaschke products one must first understand its definition. The Blaschke condition as stated in (0.2) is a condition of convergence. It says that if (0.2) converges then so will (0.1). Therefore, the true definition of a Blaschke product comes from the Blaschke condition since if it is true for a sequence of 𝑎𝑛 then so is (0.1) for the same sequence. To prove the connection between the conditions we first study (0.1)
|𝑎𝑛| 𝑎𝑛
𝑎𝑛−𝑧
1−𝑎̅̅̅̅𝑧𝑛 = |𝑎𝑛| 𝑧−𝑎𝑛
|𝑎𝑛|2𝑧−𝑎𝑛 = (1 − (1 − |𝑎𝑛|)) (1 + 𝑧(1+|𝑎𝑛|)
|𝑎𝑛|2(𝑧−1
𝑎𝑛)(1 − |𝑎𝑛|)) (1.1) Since 𝑧(1+|𝑎𝑛|)
|𝑎𝑛|2(𝑧−1
𝑎𝑛)≤ 6|𝑧|
1−|𝑧| the convergence of (1.1) can be said to be definite if |𝑧| < 1 and
∑ (1 − |𝑎𝑛 𝑛|) < ∞ and thereby the Blaschke condition is proven. The factor 𝑒𝑖𝑎 can be ignored with regards to convergence since it only rotates the function.
1.2 Finite Blaschke products
This paper will focus on the study of finite Blaschke products meaning that (0.1) will be rewritten
𝐵(𝑧) = 𝑒𝑖𝑎𝑧𝑘∏ |𝑎𝑛|
𝑎𝑛 𝑎𝑛−𝑧 1−𝑎̅̅̅̅𝑧𝑛
𝑚𝑛=1 = 𝑒𝑖𝑎𝑧𝑘∏𝑚𝑛=1𝑏𝑎𝑛(𝑧) (1.2) Where 𝑘 ∈ ℕ0, 𝑎 ∈ ℝ and {𝑎1, 𝑎2, … , 𝑎𝑚} is a finite set on the unit disc or in other words they have a finite number of zeros on the unit disc, in this formulation m if you count with multiples. 𝑏𝑎𝑛(𝑧) is called the Blaschke factor. Since the degree of a rational fraction is max(deg(nominator),deg(denominator)) the degree of a finite Blaschke product is m.
Because of the formulation of the Blaschke factor the Blaschke products have some interesting and useful characteristics. One important of these characteristics is
automorphism. As stated by Garcia, Mashreghi and Ross the automorphism comes as a consequence of the infamous Schwarz Lemma.2
From the automorphism characteristic the modulus and argument of a Blaschke factor can be derived. The modulus can easily be found since
𝑏𝑎𝑛(𝑧) = |𝑎𝑎𝑛|
𝑛 𝑎𝑛−𝑧
1−𝑎̅̅̅̅𝑧𝑛 ⇒ |𝑏𝑎𝑛(𝑧)|2 = 1 −(1−|𝑧|2)(1−|𝑎𝑛|2)
|1−𝑎̅̅̅̅𝑧|𝑛 2 (1.3)
2 S. Garcia, J. Mashreghi, W. Ross; “Finite Blaschke products: A survey”, (2015)
5
According to Garcia, Mashreghi and Ross the argument on the edge of the unit circle can be derived from:
𝑏𝑎𝑛(𝑒𝑖𝜃) = −𝑒𝑖(𝜃−𝜃𝑛) 1−𝑟𝑛𝑒−𝑖(𝜃−𝜃𝑛)
1−𝑟𝑛𝑒𝑖(𝜃−𝜃𝑛 = 𝑒𝑖𝑎𝑟𝑔𝑏𝑎𝑛(𝑒𝑖𝜃) (1.4) In which −𝜋 ≤ 𝑎𝑟𝑔𝑏𝑎𝑛(𝑒𝑖𝜃) < 𝜋. Then we can write the argument as.
𝑎𝑟𝑔𝑏𝑎𝑛(𝑒𝑖𝜃) = {
−𝜋
−2 arctan ( 1−𝑟𝑛
(1+𝑟𝑛) 𝑡𝑎𝑛(𝜃−𝜃𝑛2 )) 𝑖𝑓 𝜃∈(𝜃𝑖𝑓 𝜃=𝜃𝑛,𝜃𝑛𝑛+2𝜋) (1.5) This only holds under the assumption that −𝜋 ≤ 𝑎𝑟𝑔𝑏𝑎𝑛(𝑒𝑖𝜃) < 𝜋 and that the zero is written as 𝑎𝑛 = 𝑟𝑛𝑒𝑖𝜃𝑛. The calculation for the argument inside the unit circle and stretch beyond the scope of this paper. However, Garcia, Mashreghi and Ross conclude it as:3
𝑎𝑟𝑔𝑏𝑎𝑛(𝑧) = arcsin|𝑎𝐼𝑚(𝑎𝑛𝑧̅)(1−|𝑎𝑛|2)
𝑛||𝑎𝑛−𝑧||1−𝑎̅̅̅̅𝑧|𝑛 (1.6) From the modulus and the argument one can derive some important insights:
|𝐵(𝑧)| < 1, 𝑖𝑓 𝑧 ∈ 𝐷 (1.7)
Since the Blaschke product comes from multiplying the Blaschke factors the modulus of the Blaschke product equals product of the moduli of the Blaschke factors. From (1.3) we get the modulus of the Blaschke factor and
𝑧 ∈ 𝐷 ⇒ |𝑧| < 1 ⇒ 0 <(1−|𝑧|2)(1−|𝑎𝑛|2)
|1−𝑎̅̅̅̅𝑧|𝑛 2 < 1 ⇒ {(1.3)} ⇒ |𝑏𝑎𝑛(𝑧)|2 < 1 ⇒ |𝐵(𝑧)| < 1 .
□
|𝐵(𝜁)| = 1, 𝑖𝑓 𝜁 ∈ 𝑇 (1.8)
By the same reasoning as for (1.7) and since 𝑧 ∈ 𝑇 ⇒ |𝑧| = 1, (1−|𝑧||1−𝑎2)(1−|𝑎𝑛|2)
̅̅̅̅𝑧|𝑛 2 = 0 ⇒
|𝑏𝑎𝑛(𝑧)|2 = 1 ⇒ |𝐵(𝑧)| = 1
□
|(𝐵(𝑧)| > 1, 𝑖𝑓 𝑧 ∈ ℂ\𝐷 (1.9) 𝑧 ∈ ℂ\𝐷 ⇒ |𝑧| > 1 ⇒(1−|𝑧||1−𝑎2)(1−|𝑎𝑛|2)
̅̅̅̅𝑧|𝑛 2 < 1
□
𝐵(𝑧) = 1
𝐵(1𝑧̅)
̅̅̅̅̅̅̅ (1.10)
1 𝐵(1𝑧̅)
̅̅̅̅̅̅̅= 1
𝑒𝑖𝑎𝑧𝑘∏ |𝑎𝑛|
𝑎𝑛 𝑎𝑛−1
𝑧̅
1−𝑎𝑛̅̅̅̅1 𝑧̅
𝑚𝑛=1
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅= 𝑒𝑖𝑎𝑧𝑘 1
∏ |𝑎𝑛|
𝑎𝑛
̅̅̅̅
𝑚𝑛=1
∗ 1
∏ 𝑎𝑛̅̅̅̅−
1 𝑧 1−𝑎𝑛1
𝑧 𝑚𝑛=1
= 𝑒𝑖𝑎𝑧𝑘∏ 𝑎̅̅̅̅𝑛
|𝑎𝑛| 1−𝑎𝑛1𝑧
𝑎𝑛
̅̅̅̅−1 𝑧
𝑚𝑛=1 =
𝑒𝑖𝑎𝑧𝑘∏ |𝑎𝑛|𝑎̅̅̅̅𝑛
|𝑎𝑛|∗|𝑎𝑛| 𝑎𝑛−𝑧 1−𝑎̅̅̅̅𝑧𝑛
𝑚𝑛=1 = 𝑒𝑖𝑎𝑧𝑘∏ |𝑎𝑛|𝑎̅̅̅̅𝑛
𝑎𝑛
̅̅̅̅𝑎𝑛 𝑎𝑛−𝑧 1−𝑎̅̅̅̅𝑧𝑛
𝑚𝑛=1 = 𝐵(𝑧)
□
3 Ibid.
6
All finite Blaschke products have these properties and they are very useful to
understand the behaviour and limitations of Blaschke products but their usefulness in applications largely comes from the fact that a composite of two Blaschke products also is a Blaschke product.4
1.3 Critical points
The critical points of a Blaschke product are defined as a sequence of points (𝜉𝑛) on the unit circle that are the zero-set of the derivative of the Blaschke product. The theory and writings of the critical sets of analytic functions such as Blaschke products is vast and most of it is beyond the scope of this paper. The focus in this section will mainly be on the basic
characteristics of the critical sets and their connections to the zero sets of Blaschke products.
The study of the critical points of the Blacshke products must begin in the spaces that the derivatives belong to. The perhaps most important of these is the weighted Bergman space 𝐴𝑝,𝛼 with 0 < 𝑝 < ∞ and 𝛼 > −1 that consists of all holomorphic function on the unit disc for which
‖𝑓‖𝑝,𝛼𝑝 = ∫ ∫ |𝑓(𝑟𝑒01 02𝜋 𝑖𝜃)|𝑝(1 − 𝑟)𝛼𝑑𝜃𝑑𝑟 (1.11) is finite. The derivative of a finite Blaschke product (𝐵′) belongs to a weighted Berman space 𝐴𝑝,𝛼 if and only if it fulfils the requirements that:
1. 0 < 1 + 𝛼 < 𝑝 < ∞ (1.12) 2. ∫ (1 − 𝑟)𝛼∫ (1−|𝐵(𝑟𝑒𝑖𝜃)|
1−𝑟 )
𝑝 2𝜋 𝑑𝜃𝑑𝑟
0 < ∞
1
0 (1.13)5
According to Hong Oh Kim in his paper “Derivatives of Blaschke Products” if 𝐵′∈ 𝐴𝑝,𝑝−2 for any p > 1 then B is a finite Blaschke product.6 To prove this theorem, we first suppose that B(z) is an infinite Blaschke product. Let 0 < 𝜀 < 1 be fixed and 𝑧𝑛 = 𝑟𝑛𝑒𝑖𝜃𝑛, 𝑧 = 𝑟𝑒𝑖𝜃 then
|𝐵(𝑧)| < 𝜀 if |𝑧𝑛− 𝑧| < 𝜀(1 − 𝑟𝑛 ) since |𝐵(𝑧)| ≤|𝑎𝑛−𝑧|
1−|𝑎𝑛| by the triangle inequality.
|𝑎𝑛− 𝑧|2 = (𝑟 − 𝑟𝑛)2+ 4𝑟𝑛𝑟𝑠𝑖𝑛2(𝜃−𝜃𝑛
2 ) ≤ (𝑟 − 𝑟𝑛)2+ (𝜃 − 𝜃𝑛)2 ⇒ If 𝑟𝑛 ≤ 𝑟 < 𝑟𝑘+𝜀(1−𝑟𝑛)
2 then |𝑧𝑛− 𝑧|2 ≤𝜀2(1−𝑟𝑛)2
4 + (𝜃 − 𝜃𝑛)2 which means that
|𝐵(𝑧)| < 𝜀 if 𝑟𝑛 ≤ 𝑟 < 𝑟𝑘+𝜀(1−𝑟𝑛)
2 and |𝜃 − 𝜃𝑛| <√3
2 𝜀(1 − 𝑟𝑛). By applying these results on (1.13) we get that for an infinite Blaschke product
4 Ibid.
5 P. R. Ahern, “The Poisson integral of a singular measure”, (1979)
6 H. O. Kim, “Derivatives of Blaschke products”, (1984)
7
∫ (1 − 𝑟)𝑝−2∫ (1−|𝐵(𝑟𝑒𝑖𝜃)|
1−𝑟 )
𝑝 2𝜋 𝑑𝜃𝑑𝑟
0 1
0 ≥ √3𝜀(1 − 𝜀)𝑝(1 − 𝑟𝑛) ∫𝑟𝑟𝑛+𝜀(1−𝑟𝑛)2 (1 − 𝑟)−𝑝(1 −
𝑛
𝑟)𝑝−2𝑑𝑟 =√3𝜀2(1−𝜀)𝑝
2−𝜀
If the sequence of zeros is instead substituted for an infinite subsequence{𝑧𝑛𝑖} that fulfil
|𝑧𝑛𝑖| +𝜀(1−|𝑧𝑛𝑖|)
2 < |𝑧𝑛𝑖+1|, 𝑖 = 1,2, …
And make out the zero set for new Blaschke product 𝐵𝑖(𝑧).
∫ (1 − 𝑟)𝑝−2∫ (1−|𝐵(𝑟𝑒𝑖𝜃)|
1−𝑟 )
𝑝 2𝜋 𝑑𝜃𝑑𝑟
0 1
0 ≥ ∑ ∫𝑟𝑟𝑛𝑖+𝜀(1−𝑟𝑛𝑖)2 (1 −
𝑛𝑖
∞𝑖=1
𝑟)𝑝−2∫ (1−|𝐵𝑖(𝑟𝑒𝑖𝜃)|
1−𝑟 )
𝑝 2𝜋 𝑑𝜃𝑑𝑟
0 ≥ ∑ √3𝜀2(1−𝜀)𝑝
2−𝜀
∞𝑖=1 → ∞
Because of this 𝐵′∉ 𝐴𝑝,𝑝−2 for an infinite Blaschke product which means that if 𝐵′∈ 𝐴𝑝,𝑝−2
then B must be finite.
□
The actual expressions for the derivative of the Blaschke product is quite easily derived and for a finite Blaschke product it can be written as a finite sum:
𝐵′(𝑧) = ∑ 𝐵(𝑧)(1−𝑎𝑎̅̅̅̅𝑧)𝑛
𝑛−𝑧
(1−|𝑎𝑛|2) (1−𝑎̅̅̅̅𝑧)𝑛 2
𝑛 (1.14)
The fact that the derivative of a finite Blaschke product leads to one of the most important theorems of this paper. As presented by Daniela Kraus in her paper “Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature” if you let 𝜉𝑗 be a sequence on the unit circle then the following statements are equivalent:
1. There exists an analytic self-map of the unit circle with critical set (𝜉𝑗).
2. There exists an indestructible Blaschke product with critical set (𝜉𝑗).
3. There exists a function in the weighted Bergman space 𝐴𝑝,𝑝−2 with zero set (𝜉𝑗).
The function that the third statement is referring to is the derivative of the Blaschke function and something we proved earlier. The equivalence of that statement with the second is of importance since it means that not just from a prescribed Blaschke product can you prove the existence of a derivative in the weighted Bergman space but that the opposite is true as well, or as Gunter Semmler and Elias Wegert put it:
“Let 𝜉1, … , 𝜉𝑛−1 be n−1 points in the unit circle. Then there is a Blaschke product B of degree n with critical points 𝜉𝑛. B is unique up to post-composition with a conformal automorphism
of D.” 7 (1)
This theorem was first proven by Maurice Heins and can be formulized in many ways. For example one interesting consequence of the uniqueness part of the theorem is that two holomorphic maps (f,g) only have the same critical points if 𝑓 = 𝜏 ∘ 𝑔 where 𝜏 is a conformal
7 G. Semmler, E. Wegert, “Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problem”, (2017)
8
automorphism of the unit disk. This equivalence can be understood by studying the fact that for any two points there exists a unique conformal automorphism which maps the points to 0 and 1 respectively. This characteristic means that a composite between the conformal automorphism and a Blaschke product will still fulfil the Blaschke condition. Since a conformal mapping does by definition not have any non-zero solution to its derivative the critical set will be the same for the composite: 𝑓′= 𝜏′(𝑔) ∗ 𝑔′
Figure 1.
8
The picture shows an example of a Blaschke product with zeros in the white dots and critical points in the grey ones.
8 “Blaschke Tool User's Guide.” English Word Frequency Lists * Lexiteria Corporation, lexiteria.com/~ashaffer/blaschke/guide/guide.html.
9
2. The Problem and its Analytical Solution
The theoretical background now leads us to the actual problem of this paper, the problem of determining and computing a Blaschke product from its prescribed critical values. In this chapter I will present two ways forward: The method described by Daniela Kraus and Oliver Roth that depend on the theory of constant gaussian curvature and equivalating it to other problems in order to provide different viewpoints.
2.1 Gaussian curvature
Gaussian curvature is an intrinsic property of a space and defined pointwise (p) as 𝐾(𝑝) = det (𝑆(𝑝)) where S is the shape operator defined as the negative derivative of the unit normal vector field of the surface. Gaussian curvature is therefore a measurement of total curvature of the surface, the most intuitive way of seeing it is by writing it as the product of the principal curvature of the surface. However, for the purpose of this paper we limit ourselves to the gaussian curvature 𝜅𝜆 of a regular conformal pseudometric 𝜆(𝑧)|𝑑𝑧| on G where G is simply connected domain in the complex plane.
𝜅𝜆(𝑧) ∶=∆(log 𝜆)(𝑧)
𝜆(𝑧)2 (2.1)
When 𝑧 ∈ 𝐺 and 𝜆(𝑧) > 0.
If the regular conformal pseudometric has a predetermined gaussian curvature, the function 𝑢 ∶= log 𝜆 satisfies the partial differential equation called the Gauss curvature equation:
∆𝑢 = −𝜅(𝑧)𝑒2𝑢 (2.2)
For many problems it is useful to simplify the gaussian curvature and only study constant curvature. This is because of the implications from a famous theorem called Liouville’s Theorem:
“Let G be a simply connected domain and let ℎ: 𝐺 → 𝐶 be an analytic map. If 𝜆(𝑧)|𝑑𝑧| is a regular conformal metric with curvature −4|ℎ(𝑧)|2 then there exists a holomorphic function 𝑓: 𝐺→𝐷 such that
𝜆(𝑧) = 1
|ℎ(𝑧)|
|𝑓′(𝑧)|
1−|𝑓(𝑧)|2, 𝑧 ∈ 𝐺 (2.3)
Moreover, f is uniquely determined up to postcomposition with a unit disk automorphism.”9 (2)
The importance of this theorem comes the insight that a conformal pseudometric with constant curvature −4 can be represented by a holomorphic function. This means that when the gaussian curvature is −4 and G is a simply connected domain there is a connection
9 D. Kraus, O. Roth, “Critical points, the gauss curvature equation & Balschke products”, (2013)
10
between critical sets of the holomorphic function and the zero-set of the corresponding conformal pseudometric.
By using this connection, the following method can be derived:
Let h be a holomorphic function that maps from the unit circle to the whole complex plane with a zero set C. Then the following statements are equivalent.
1. There exists a holomorphic function f : 𝐷 → 𝐷 with critical set C.
2. There exists a 𝐶2-solution u : 𝐷 → 𝑅 to the Gauss curvature equation
∆𝑢 = 4|ℎ(𝑧)|2𝑒2𝑢 (2.4)
To prove this we begin by computing the corresponding conformal pseudometric with (2.3) and then creating the function 𝑢 = log(𝜆(𝑧)) = log (|ℎ(𝑧)|1 |𝑓′(𝑧)|
1−|𝑓(𝑧)|2). Since both f(z) and h(z) belongs to the unit circle and their moduli belongs to ℝ it is obvious that u will map from the unit circle to ℝ.
∆𝑢 = 𝜕2
𝜕𝑧𝜕𝑧̅(log ( 1
|ℎ(𝑧)|
|𝑓′(𝑧)|
1 − |𝑓(𝑧)|2)) =
= 4|ℎ(𝑧)|2(|ℎ(𝑧)|1 |𝑓′(𝑧)|
1−|𝑓(𝑧)|2)
2
And has seen above u is also a solution to (2.4) which ends the proof of 1. → 2. To prove the reverse we draw the conclusion that if there exists a 𝐶2-solution to (2.4) then 𝜆(𝑧)|𝑑𝑧| is a regular conformal pseudometric which by (2) means that there exists a holomorphic f such that it satisfies (2.3). Since G is here defined as the unit circle the proof is complete. 10
□
This equivalency can now be used to determine a Blaschke product f from its critical values C. This is because of the uniqueness theorem (1). The solution is quite elegant, but the obvious problem is the necessity of a predetermined gauss curvature without which the equivalency does not exists. Another restriction is that G as defined in (2) must be a simply connected domain. However, this does not cause much worries for our problem since the unit circle on which Blaschke-products are defined is a simply connected domain.
10 Ibid.
11
2.2 Equivalence to other problems
In this section we will take a different approach to the problem and prove the equivalency to a different type of probles. The goal is to widen the readers view of Blaschke products and their applications and to give different ways forward in studying the problem.
2.2.1 Equilibrium position of point charge
The problem of determining a Blaschke product from its critical value is equivalent to the problem of finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges. To prove this assertion, we transform the original problem to a second order ordinary differential equation on 𝐻 ∶= {𝑥 + 𝑖𝑦, 𝑦 > 0}. To do this we use the Cayley transform
𝑇(𝑧) = 𝑖1+𝑧
1−𝑧 (2.5)
If we let B be a Blaschke product that fulfil the criteria 𝐵(1) = −1 than we can now create a rational function that is real-valued on ℝ with corresponding critical values to B, because of the form of (2.6) a critical value 𝜉𝑛 of B is only that if 𝜁𝑛 = 𝑇(𝜉𝑛) is a critical value of f.
𝑓 ∶= 𝑇 ∘ 𝐵 ∘ 𝑇−1 (2.6)
As can be seen though is that the function f is not real-valued at its poles. Because of this and the fact that it is a rational function we can write it as a fraction decomposition:
𝑓(𝑥) = − 𝑟1
𝑥−𝑥1− 𝑟2
𝑥−𝑥2− ⋯ − 𝑟𝑛
𝑥−𝑥𝑛 (2.7)
Where 𝑥𝑛 are the poles with 𝑥1 < 𝑥2 < 𝑥3 < ⋯ < 𝑥𝑛 and 𝑟𝑛 are constants and > 0.
𝑓′(𝑥) =𝑥−𝑥𝑟1
12+ 𝑟2
(𝑥−𝑥2)2+ ⋯ + 𝑟𝑛
(𝑥−𝑥𝑛)2 = 𝑐𝑃(𝑥)
𝑄(𝑥)2 (2.8) The problem of determining a Blaschke product from its critical values is now reduced to determining f from its critical values since T is known and critical values correspond with each other.
This problem can be written as an ODE where the goal is to find a real polynomial (R) that satisfy the equation:
𝑃𝑄′′− 𝑃′𝑄′+ 𝑅𝑄 = 0 (2.9)
Where R must be of the degree 2n – 4, n is the degree of the Blaschke product.
This can be proven by first evaluating Q(x) and P(x).
𝑄(𝑥) = ∏𝑛𝑘=1(𝑥 − 𝑥𝑘) (2.10)
𝑃(𝑥) = ∏𝑛−1𝑘=1(𝑥 − 𝜁𝑘)(𝑥 − 𝜁̅ )𝑘 (2.11) Where 𝜁𝑘 are the critical points of f.
From (2.7) we can also see that
12
𝑐𝑃(𝑥) = ∑ 𝑟𝑘∏𝑛𝑗=1(𝑥 − 𝑥𝑗)2
𝑗≠𝑘
𝑛𝑘=1 (2.12)
By inserting (2.12) into (2.11) we get 𝑃(𝑥) = ∑ 𝑃(𝑥𝑘) ∏ (𝑥−𝑥𝑗)
2
(𝑥𝑘−𝑥𝑗)2 𝑛𝑗=1
𝑗≠𝑘
= {𝑄(𝑥) = ∏𝑛𝑘=1(𝑥 − 𝑥𝑘) , 𝑄′(𝑥) = ∏𝑛𝑘=1(𝑥 − 𝑥𝑘)2}
𝑛𝑘=1 =
∑ 𝑃(𝑥𝑘) ( 𝑄(𝑥)
𝑄′(𝑥𝑘)(𝑥−𝑥𝑘))2
𝑛𝑘=1 , 𝑘 = 1,2,3 … 𝑛. (2.13)
According to Semmler and Wegert we can then go via the fundamental polynomials of the first and second kind of Hermite interpolation to also get:
𝑃(𝑋) = ∑ 𝑃(𝑥𝑘) (𝑄′ 𝑄(𝑥)
(𝑥𝑘)(𝑥−𝑥𝑘))2
𝑛𝑘=1 + ∑ (𝑃′(𝑥𝑘) − 𝑃(𝑥𝑘)𝑄′′(𝑥𝑘)
𝑄′(𝑥𝑘) )
𝑛𝑘=1 (𝑥 − 𝑥𝑘( 𝑄(𝑥)
𝑄′(𝑥𝑘)(𝑥−𝑥𝑘))2 (2.15)
(2.14) must equal (2.15) and since (𝑥 − 𝑥𝑘) ( 𝑄(𝑥)
𝑄′(𝑥𝑘)(𝑥−𝑥𝑘))2are linearly independent, meaning they can not be expressed as factors of each other and we need only study 𝑃′(𝑥𝑘) −
𝑃(𝑥𝑘)𝑄′′(𝑥𝑘)
𝑄′(𝑥𝑘). So for the equations to be the same 𝑃′(𝑥𝑘) − 𝑃(𝑥𝑘)𝑄′′(𝑥𝑘)
𝑄′(𝑥𝑘) = 0. When
studying this we can see that all of them depend on 𝑥𝑘 and must equal 0 for all 𝑥𝑘. This is of course also true of Q for which 𝑥𝑘 are the roots. Therefore, can we write
𝑃𝑄′′− 𝑃′𝑄′= −𝑅𝑄 (2.14)
Where R is some real polynomial.
□
The interpretation of this is that the function f has the prescribed critical points if and only if there is a real polynomial R that fulfil (2.14). The problem then becomes to determine the functions P(x) and Q(x) so that we can determine f and in extension B. The process of doing this is beyond the scope of this paper, however an interesting interpretation comes from this process which I will just present. According to Semmler and Wegert to solve (2.14) one must satisfy the equation
∑ 2
𝑥𝑘−𝑥𝑗 𝑛𝑗=1 𝑗≠𝑘
− ∑ ( 1
𝑥𝑘−𝜁𝑗+ 1
𝑥𝑘−𝜁̅𝑗) = 0 , 𝑘 = 1,2,3 … , 𝑛
𝑛−1𝑗=1 (2.15)
Equation (2.15) has a physical interpretation. There are fixed charges with q = -1/2 at the points 𝜁𝑗 and 𝜁̅ . At the poles 𝑥𝑗 𝑘 are n positive unit charges that are in equilibrium in the field created by the all the charges. Therefore, by studying the equilibrium moveable charges one could solve the problem of determining a Blaschke product from its critical values.11
11 G. Semmler, E. Wegert, “Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems”, (2017)
13
3. Discrete Solutions
While there has not been any analytical solution to the problem of determining a Blaschke product from its critical values there are some discrete methods. This paper will present two of them, one that is based on the theory of circle packing where discrete Blaschke products are locally approximated as circle packing maps and one that rewrites the problem as a quadratic system of equations and solves it numerically.
3.1 Circle packing
Circle packings are defined as a group of circles of either equal or varying size that follow a prescribed pattern, an example of this can be seen in Figure 2. A circle packing must also have a one-to-one relationship between any potential vertices of the body it is filling and a circle. Vertices that are connected by an edge as in for example a square or triangle the corresponding circles of these vertices must be externally tangent. A circle packing must also be orientation preserving. The circle packing is said to be univalent if all the circles are mutually disjoint and furthermore if the univalent circle packing is filling the unit circle is called Andreev packings.
Figure 2.
Figure 2. shows a geometrical representation of a univalent circle packing in a hexagonal arrangement12
12 “Circle Packing.” Wikipedia, Wikimedia Foundation, 10 Apr. 2018, en.wikipedia.org/wiki/Circle_packing.
14
One important characteristic of circle packings is branching. A branch point of a circle packing is a point of discontinuity and are caused by vertices in the filled body. If a circle packing contains a branch point it is said to be a branched circle packing. Each branch point will also have an angle sum which for circle packings always is 2𝑛𝜋, 𝑛 ≥, where n is called the multiplicity of P. Every branched circle packing will therefor have a set of paired vertices and multiplicities. This set is called a branch set of the circle packing. The branched versions of Andreev packings are usually called Bl-type packing.
To use branched circle packings to approximate finite Blaschke product we will first use two theorems presented by Tomasz Dubejko:
1. The first one says that for a given body T with a given branch set ℬ =
{(𝑏1, 𝑘1), (𝑏2, 𝑘2), … , (𝑏𝑚, 𝑘𝑚)}, where 𝑏𝑚 denotes the vertices and 𝑘𝑚 denotes the multiplicity, there exists branched circle packing P if and only if ℬ is also a branch structure for T.
2. The second one is a the uniqueness-theorem for the first one and it says that if you let T be a finite triangulation of a disc and you suppose ℬ =
{(𝑏1, 𝑘1), (𝑏2, 𝑘2), … , (𝑏𝑚, 𝑘𝑚)} is a branch structure for T then the existing branched circle packing as determined by T is unique up to isometries in the corresponding geometry.13
The proofs for these theorems are quite demanding and time-consuming and I have
therefore deemed them outside the scope of this paper but for the interested reader I refer you to Tomasz Dubejkos article.
The idea of a discrete Blaschke product is that it maps from Andreev packings to Bl-type packings. This can be better understood by studying a map f from an Andreev packing A and a Bl-type packing B. Since both packings are defined on the unit circle and by Stoilow’s theorem in topology you can write a circle mapping as a composite between a
homeomorphism, ℎ, and a holomorphic function, 𝜑, within the homeomorphism ℎ(𝐷).
𝑓 = 𝜑 ∘ ℎ (3.1)
h(D) is per definition simply connected and by the Riemann mapping theorem that states that for every simply connected open subset of the complex plane there exist a holomorphic function that maps onto the unit circle. This means that the homeomorphism can be
transformed such that ℎ: 𝐷 ⟶ 𝐷 which means that 𝜑: 𝐷 → 𝐷. 𝜑 is now an analytical
function and automorphism which means it can be written as a finite Blaschke product. This leads to the mathematical definition of a discrete Blaschke product. A mapping f from an Andreev circle packing to a Bl-type packing for a prescribed T with a set branch structure is called a discrete Blaschke product. f can be written as a composite between a
homeomorphism h: D -> D and a finite Blaschke product.
13 T. Dubejko, ”Branched Circle Packings and Discrete Blaschke Products”, (1995)
15
According to Tomasz Dubejko we can use these discrete Blaschke product to approximate a finite Blaschke product. Because for every finite Blaschke product there exists a sequence of discrete Blaschke products that uniformly converges on the unit circle.
The reason we can use this result is because the branch points of the Bl-type packing will correspond with the critical values of the approximated finite Blaschke product. Therefore, by choosing the branch points we can determine a Blaschke product from its prescribed critical points.
3.2 A quadratic model
The method of using a quadratic model to numerically compute a Blaschke product from its critical values begins in writing a Blaschke product as a rational function of complex
polynomials:
𝐵(𝑧) =𝑝(𝑧)
𝑞(𝑧)= 𝑎0+𝑎1𝑧+⋯+𝑎𝑛𝑧𝑛
𝑎𝑛
̅̅̅̅+𝑎̅̅̅̅̅̅̅𝑧+⋯+𝑎𝑛−1 ̅̅̅̅𝑧0 𝑛 (3.2) Where 𝑎𝑖 ∈ 𝐶 and p(z) must have all its zero in the unit circle.
The goal is of course to determine B by solving the equations
𝐵′(𝑧𝑗) = 0, 𝑗 = 1,2, … , 𝑛 (3.3) (3.3) can now be rewritten by deriving 𝑝(𝑧)𝑞(𝑧) to create a system of equations
𝐵′(𝑧𝑗) = 0 ⇔ 𝑝′(𝑧𝑗)𝑞(𝑧𝑗) − 𝑝(𝑧𝑗)𝑞′(𝑧𝑗) = 0 ⇔ ∑𝑛+1𝑖=1 ∑𝑛𝑗=1(𝑖 − 𝑗)𝑎𝑖𝑎̅𝑛−𝑗+1𝑧𝑘𝑖+𝑗−1+
∑𝑛𝑖=1𝑖𝑎𝑖𝑧𝑘𝑖−1+ (𝑛 + 1)𝑧𝑘𝑛 = 0 (3.4)
Where 𝑧𝑘 are the set of the prescribed critical values and their inverted conjugates because 𝐵′(𝑧𝑗) = 0 ⇒ 𝐵′(1
𝑧𝑗
̅̅̅).
(3.4) is now numerically solvable, however according to the paper of Ray Pörn and Christer Glader this system generates very high conditional numbers. They therefore recommend that you transform the equations first to matrix form 𝐴𝑥 = 𝑏 where x is the sought-after solution vector. Then take one particular solution, 𝛼, to this as well as storing the basis for the null space of A in a new vector, 𝑉, to create a model: 𝑥 = 𝛼 + 𝑉𝑑 where d is a constant that belongs to the complex plane. When you solve this numerically 𝛼 = 𝑥0 is used as a starting point. The method unfortunately generates lot of possible solution and according to (1) is there one unique solution to the problem. Ray Pörn and Christer Glader solves this by stating that the Blaschke product corresponds to the solution with the smallest value of |𝑎1|.
14
14 R. Pörn, C. Glader, “On the construction of finite Blacshke products with prescribed critical points”, (2011)
16 Figure 3.
Figure 3. shows a Blaschke product generated by the quadratic model, the black spots represent critical values and the white spots represent zeros.15
15 Ibid.
17
References
Articles
W. Blaschke, "Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen" Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig , 67 (1915)
S. Garcia, J. Mashreghi, W. Ross; “Finite Blaschke products: A survey”, (2015) P. R. Ahern, “The Poisson integral of a singular measure”, (1979)
H. O. Kim, “Derivatives of Blaschke products”, (1984)
G. Semmler, E. Wegert, “Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problem”, (2017)
D. Kraus, O. Roth, “Critical points, the gauss curvature equation & Balschke products”, (2013) T. Dubejko, “Branched Circle Packings and Discrete Blaschke Products”, (1995)
R. Pörn, C. Glader, “On the construction of finite Blacshke products with prescribed critical points”, (2011)
Websites
“Blaschke Tool User's Guide.” English Word Frequency Lists * Lexiteria Corporation, lexiteria.com/~ashaffer/blaschke/guide/guide.html.
“Circle Packing.” Wikipedia, Wikimedia Foundation, 10 Apr. 2018, en.wikipedia.org/wiki/Circle_packing.
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