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, Chebyshev-Blaschke 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. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (M.Sc.(Mathematics and Applied Mathematics)) -- Universiteit van Pretoria, 2007. / Includes bibliographical references. Available on the Internet via the World Wide Web.

No abstract available / Dissertation (MSc (Mathematics))--University of Pretoria, 2008. / Mathematics and Applied Mathematics / unrestricted

Titre de l'écran-titre (visionné le 13 novembre 2023) / Dans les dernières années, un analogue non linéaire aux séries de Fourier a intéressé plusieurs mathématiciens. Ce dernier permet d'approximer un signal par une somme de termes dont les composantes représentent la fréquence et l'amplitude. Il s'agit du déroulement de Blaschke de fonctions analytiques introduit par Coifman, ou développement de Fourier adaptatif. L'idée de Coifman a été de factoriser toutes les racines dans le disque unité en interprétant les monômes z ↦ zⁿ présents dans la série de Taylor comme des produits de Blaschke. Il a aussi utilisé la factorisation de Blaschke pour les fonctions analytiques sur un voisinage du disque unité. Ce développement en série a été appliqué à plusieurs autres problèmes depuis, car il présente de nombreux avantages sur les séries de Fourier classiques. Néanmoins, la question de convergence de cette représentation en série est un problème majeur depuis plusieurs décennies. On sait seulement qu'il y a convergence de la série dans certains sous-espaces de H² avec poids et, par des résultats récents, dans les espaces de Hardy. Dans ce mémoire, on présente un déroulement de fonctions dans les espaces de Hilbert à noyau reproduisant et dans les espaces de Hardy qui est une généralisation du déroulement de Blaschke et qui est inspiré par la théorie des opérateurs et les espaces de de Branges-Rovnyak. Pour ce faire, on développe d'abord les notions préalables de l'analyse complexe, harmonique et fonctionnelle. Nos résultats principaux sont des théorèmes de convergence pour ces développements en série. Quelques applications et exemples sont aussi présentés. / Over the last few years, many mathematicians became interested in a nonlinear analogue of Fourier series that allows them to approximate a signal by a sum of terms whose components represent frequency and amplitude. It is the Blaschke unwinding series introduced by Coifman, or adaptive Fourier decomposition. Coifman's idea was to factor all the roots in the unit disk by thinking of the monomials z ↦ zⁿ in the Taylor series as Blaschke products. He also used the Blaschke factorization for analytic functions in a neighbourhood of the unit disk. Because it has many advantages over the classical Fourier series, this series expansion has been used in several other problems since. Yet, the question of convergence of the series has remained a major problem for a few decades. We only know that it converges in certain weighted subspaces of H² and, by recent work, in Hardy spaces. In this thesis, we introduce an expansion scheme in reproducing kernel Hilbert spaces and Hardy spaces. It is a generalization of the Blaschke unwinding series expansion which is motivated by operator theory and de Branges-Rovnyak spaces. To do this, we first introduce the necessary background material in complex analysis, harmonic analysis, and functional analysis. Our main results are convergence theorems for these series expansions. We also present some applications and examples.

A geometria de teias dedica-se ao estudo de invariantes locais para uma determinada configuração de folheações. Uma d-teia é uma coleção de folheações que estão em posição geral. Desta forma, uma d-teia plana, definida em \'R POT.2\' ou \'C POT.2\', nada mais é que uma família de d folheações por curvas. Apresentamos neste trabalho os principais conceitos da teoria clássica de teias, iniciada por W. Blaschke por volta de 1930, bem como uma abordagem atual utilizada no estudo de teias planas. São abordados dois tipos de problemas importantes na teoria: os problemas de linearização e de algebrização de teias. Provamos um resultado clássico no que concerne ao problema de linearização, e um resultado de algebrização de teias empregando métodos desenvolvidos mais recentemente / Web geometry is devoted to the study of local invariants of a certain configuration of foliations. A d-web is a collection of foliations in general position. Therefore, a d-web defined in \'R POT. 2\' or \'C POT. 2\' is just a family of d foliations by curves. We present in this work the main concepts of classical theory of webs, initiated by W. Blaschke around 1930, as well as newer methods used in the study of plane webs. We approach two important types of problems in the theory: problems of linearization and that of algebrization of webs. We prove a classical result concerning the linearization problem, and a result of algebrization of webs using recently developed methods

A geometria de teias dedica-se ao estudo de invariantes locais para uma determinada configuração de folheações. Uma d-teia é uma coleção de folheações que estão em posição geral. Desta forma, uma d-teia plana, definida em \'R POT.2\' ou \'C POT.2\', nada mais é que uma família de d folheações por curvas. Apresentamos neste trabalho os principais conceitos da teoria clássica de teias, iniciada por W. Blaschke por volta de 1930, bem como uma abordagem atual utilizada no estudo de teias planas. São abordados dois tipos de problemas importantes na teoria: os problemas de linearização e de algebrização de teias. Provamos um resultado clássico no que concerne ao problema de linearização, e um resultado de algebrização de teias empregando métodos desenvolvidos mais recentemente / Web geometry is devoted to the study of local invariants of a certain configuration of foliations. A d-web is a collection of foliations in general position. Therefore, a d-web defined in \'R POT. 2\' or \'C POT. 2\' is just a family of d foliations by curves. We present in this work the main concepts of classical theory of webs, initiated by W. Blaschke around 1930, as well as newer methods used in the study of plane webs. We approach two important types of problems in the theory: problems of linearization and that of algebrization of webs. We prove a classical result concerning the linearization problem, and a result of algebrization of webs using recently developed methods

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Thesis (Ph. D.)--West Virginia University, 2007. / Title from document title page. Document formatted into pages; contains x, 92 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 75-76).

Thèse (Ph. D.)--Université Laval, 2008. / Bibliogr.: f. [83]-85. Publié aussi en version électronique dans la Collection Mémoires et thèses électroniques.

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