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postgraduate thesis: Finite Blaschke products versus polynomials
Title  Finite Blaschke products versus polynomials 

Authors  
Advisors  Advisor(s):Ng, TW 
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Tsang, C. [曾超賢]. (2012). Finite Blaschke products versus polynomials. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4784971 
Abstract  The objective of the thesis is to compare polynomials and finite Blaschke products, and demonstrate that they share many similar properties and hence we can
establish a dictionary between these two kinds of finite maps for the first time.
The results for polynomials were reviewed first. In particular, a special kind of
polynomials was discussed, namely, Chebyshev polynomials, which can be defined
by the trigonometric cosine function cos ?. Also, a complete classification for two
polynomials sharing a set was given.
In this thesis, some analogous results for finite Blaschke products were proved.
Firstly, ChebyshevBlaschke products were introduced. They can be defined by re
placing the trigonometric cosine function cos z by the Jacobi cosine function cd(u; ? ).
They were shown to have several similar properties of Chebyshev polynomials, for
example, both of them share the same monodromy, satisfy some differential equations and solve some minimization problems. In addition, some analogous results
about two finite Blaschke products sharing a set were proved, based on Dinh's and
Pakovich's ideas.
Moreover, the density of prime polynomials was investigated in two different
ways: (i) expressing the polynomials of degree n in terms of the zeros and the leading
coefficient; (ii) expressing the polynomials of degree n in terms of the coefficients.
Also, the quantitative version of the density of composite polynomials was developed
and a density estimate on the set of composite polynomials was given. Furthermore,
some analogous results on the the density of prime Blaschke products were proved. 
Degree  Doctor of Philosophy 
Subject  Blaschke products. Polynomials. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/174529 
HKU Library Item ID  b4784971 
DC Field  Value  Language 

dc.contributor.advisor  Ng, TW   
dc.contributor.author  Tsang, Chiuyin   
dc.contributor.author  曾超賢   
dc.date.issued  2012   
dc.identifier.citation  Tsang, C. [曾超賢]. (2012). Finite Blaschke products versus polynomials. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4784971   
dc.identifier.uri  http://hdl.handle.net/10722/174529   
dc.description.abstract  The objective of the thesis is to compare polynomials and finite Blaschke products, and demonstrate that they share many similar properties and hence we can establish a dictionary between these two kinds of finite maps for the first time. The results for polynomials were reviewed first. In particular, a special kind of polynomials was discussed, namely, Chebyshev polynomials, which can be defined by the trigonometric cosine function cos ?. Also, a complete classification for two polynomials sharing a set was given. In this thesis, some analogous results for finite Blaschke products were proved. Firstly, ChebyshevBlaschke products were introduced. They can be defined by re placing the trigonometric cosine function cos z by the Jacobi cosine function cd(u; ? ). They were shown to have several similar properties of Chebyshev polynomials, for example, both of them share the same monodromy, satisfy some differential equations and solve some minimization problems. In addition, some analogous results about two finite Blaschke products sharing a set were proved, based on Dinh's and Pakovich's ideas. Moreover, the density of prime polynomials was investigated in two different ways: (i) expressing the polynomials of degree n in terms of the zeros and the leading coefficient; (ii) expressing the polynomials of degree n in terms of the coefficients. Also, the quantitative version of the density of composite polynomials was developed and a density estimate on the set of composite polynomials was given. Furthermore, some analogous results on the the density of prime Blaschke products were proved.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.source.uri  http://hub.hku.hk/bib/B4784971X   
dc.subject.lcsh  Blaschke products.   
dc.subject.lcsh  Polynomials.   
dc.title  Finite Blaschke products versus polynomials   
dc.type  PG_Thesis   
dc.identifier.hkul  b4784971   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4784971   
dc.date.hkucongregation  2012   