Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Breslau, 1918. / Cover title. Vita.

Thesis (Ph. D.)--Lehigh University, 2000. / Includes vita. Includes bibliographical references (leaves 49-50).

Qin, Guiming. / "August 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 72-74). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- B splines and B-splines surfaces --- p.5 / Chapter 1. 2 --- Box spline --- p.6 / Chapter 1. 3 --- Biorthogonal subdivision wavelets based on the lifting scheme --- p.7 / Chapter 1.4 --- Geometrically-constrained subdivision wavelets --- p.9 / Chapter 1.5 --- Contributions --- p.9 / Chapter 2 --- Explicit symbol formulae for B-splines --- p.11 / Chapter 2. 1 --- Explicit formula for a general recursion scheme --- p.11 / Chapter 2. 2 --- Explicit formulae for de Boor algorithms of B-spline curves and their derivatives --- p.14 / Chapter 2.2.1 --- Explicit computation of de Boor Algorithm for Computing B-Spline Curves --- p.14 / Chapter 2.2.2 --- Explicit computation of Derivatives of B-Spline Curves --- p.15 / Chapter 2. 3 --- Explicit power-basis matrix fomula for non-uniform B-spline curves --- p.17 / Chapter 3 --- Biorthogonal subdivision wavelets with geometric constraints --- p.23 / Chapter 3. 1 --- Primal subdivision and dual subdivision --- p.23 / Chapter 3. 2 --- Biorthogonal Loop-subdivision-based wavelets with geometric constraints for triangular meshes --- p.24 / Chapter 3.2.1 --- Loop subdivision surfaces and exact evaluation --- p.24 / Chapter 3.2.2 --- Lifting-based Loop subdivision wavelets --- p.24 / Chapter 3.2.3 --- Biorthogonal Loop-subdivision wavelets with geometric constraints --- p.26 / Chapter 3. 3 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.35 / Chapter 3.3.1 --- Catmull-Clark subdivision and Doo-Sabin subdivision surfaces --- p.35 / Chapter 3.3.1.1 --- Catmull-Clark subdivision --- p.36 / Chapter 3.3.1.2 --- Doo-Sabin subdivision --- p.37 / Chapter 3.3.2 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.38 / Chapter 3.3.2.1 --- Biorthogonal Doo-Sabin subdivision wavelets with geometric constraints --- p.38 / Chapter 3.3.2.2 --- Biorthogonal Catmull-Clark subdivision wavelets with geometric constraints --- p.44 / Chapter 4 --- Experiments and results --- p.49 / Chapter 5 --- Conclusions and future work --- p.60 / Appendix A --- p.62 / Appendix B --- p.67 / Appendix C --- p.69 / Appendix D --- p.71 / References --- p.72

Title from title page of PDF (University of Missouri--St. Louis, viewed April 5, 2010). Includes bibliographical references (p. 84-89).

Este trabalho tem como objetivo principal aplicar elementos de teoria dos conjuntos no estudo de sistemas biortogonais em espaços de Banach. Inicialmente, estudamos o Teorema de Markushevic, que garante que todo espaço de Banach separável admite um sistema biortogonal enumerável. Assim, partimos para o estudo de espaços de Banach não separáveis, mais especificamente, estudamos a existência de sistemas biortogonais não enumeráveis em espaços de Banach da forma C(K), com K compacto Hausdorff não metrizável. Nesta direção, estudamos dois teoremas devido a S. Todorcevic. O primeiro teorema nos dá condições que um compacto Hausdorff K deve satisfazer de tal modo que o respectivo espaço de Banach C(K) possua sistemas biortogonais não enumeráveis. O segundo teorema nos diz que, assumindo o Axioma de Martin, todo espaço de Banach não separável da forma C(K) possui um sistema biortogonal não enumerável. Em seguida, consideramos algumas funções cardinais definidas por P. Koszmider para espaços de Banach, associadas aos sistemas biortogonais e estudamos suas relações com funções cardinais conhecidas. Em particular, obtemos um resultado original que relaciona o peso de um espaço compacto Hausdorff K com o tamanho de tipos especiais de sistemas biortogonais em C(K), generalizando um resultado de S. Todorcevic sobre álgebras de Boole. Finalmente, construímos um espaço de Ostaszewski K usando o Princípio Diamante. O espaço K é um compacto disperso não metrizável tal que todas suas potências finitas são hereditariamente separáveis. Este espaço é um exemplo consistente de um espaço compacto Hausdorff não metrizável tal que o respectivo espaço de Banach C(K) não admite sistemas biortogonais não enumeráveis. / The main purpose of this work is to apply elements of set theory to the study of biorthogonal systems in Banach spaces. Initially, we study Markushevic\'s Theorem, which ensures that every separable Banach space has a countable biorthogonal system. With this result, we focus our attention to the study of nonseparable Banach spaces, more especifically, we study the existence of uncountable biorthogonal systems in Banach spaces of the form C(K), with K a nonmetrizable compact Hausdorff space. In this direction, we study two theorems of S. Todorcevic. The first one gives us sufficient conditions that a compact Hausdorff space K must satisfy in order to get that the respective Banach space C(K) has an uncountable biorthogonal system. The second one tells us that under Martin\'s Axiom, every nonseparable Banach space of the form C(K) has an uncountable biorthogonal system. Next, we consider some cardinal functions defined by P. Koszmider for Banach spaces, related with biorthogonal systems, and we study its relations with well - known cardinal functions. In particular, we obtain an original result relating the weight of a compact Hausdorff space K to the size of certain biorthogonal systems in C(K), generalizing a result of S. Todorcevic for Boolean algebras. Finally, we construct an Ostaszewski space K using the Diamond Principle. The compact space K is a scattered nonmetrizable Hausdorff space such that all its finite powers are hereditarily separable. This space is a consistent example of a nonmetrizable compact Hausdorff space such that the respective Banach space C(K) does not have an uncountable biorthogonal system.