Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaves [141]-148).

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Matemática e Computação Científica. / Made available in DSpace on 2012-10-23T07:18:50Z (GMT). No. of bitstreams: 1 235906.pdf: 730825 bytes, checksum: 25d5e053cb093d9fd481ef9ec6be7b74 (MD5) / Neste trabalho, apresentamos resultados de estabilidade e análise de convergência dos métodos Chebyshev-espectrais para equações diferenciais parciais parabólicas. Abordamos a teoria dos métodos Fourier-espectrais considerando apenas os resultados necessários ao desenvolvimento da teoria dos métodos Chebyshev-espectrais. A existência e unicidade de soluções foram obtidas através do método Faedo-Galerkin. Estabelecemos resultados de estabilidade e convergência de esquemas semi-discretos e totalmente discretos para as equações de advecção-difusão (uni e bidimensional) e do calor bidimensional. No caso de esquemas totalmente discretos, utilizamos o método implícito teta, com teta entre 1/2 e 1, para avançar no tempo. A taxa de convergência é espectral com relação ao espaço e polinomial no tempo (segunda ordem para teta pertencente a (1/2,1] e quarta ordem para teta=1/2).

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Thesis submitted 1970; degree awarded 1971. / In this dissertation we consider the minimax approximation of functions f(x) E"C[O, l] rotated about the origin, and the characterization of the optimal rotation, a*, of f in the sense of least minimax error over all possible rotations. The paper divides naturally into two sections: a) Existence, uniqueness, and characterization for unisolvent minimax approximation for each rotation a of f. These results are applications of Dunham (1967). b) Existence, non-uniqueness, and com.putation of a*; derivation of necessary conditions for the minimax [TRUNCATED]

The purpose of this paper is to investigate the nature and practical uses of Chebyshev polynomials. Chapter I gives recognition to mathematicians responsible for studies in this area. Chapter II enumerates several mathematical situations in which the polynomials naturally arise and suggests reasons for the pursuance of their study. Chapter III includes: Chebyshev polynomials as related to "best" polynomial approximation, Chebyshev series, and methods of producing polynomial approximations to continuous functions. Chapter IV discusses the use of Chebyshev polynomials to solve certain differential equations and Chebyshev-Gauss quadrature.

Les réseaux de Chebyshev sont des systèmes de coordonnées sur les surfaces que l'on obtient par cisaillement d'un domaine du plan. Ceux-ci sont utilisés en particulier pour modéliser les gridshells qui constituent une construction architecturale notamment reconnue pour son faible coût environnemental. La difficulté principale dans la conception des gridshells est le manque de diversité des formes accessibles. En effet, bien que toute surface admette localement en tout point un réseau de Chebyshev, l'existence globale de ce type de coordonnées n'est possible que sur un ensemble restreint de surfaces. La recherche de conditions suffisantes pour l'existence globale de réseaux de Chebyshev est toujours d'actualité. Un des résultats de cette thèse est l'amélioration de ces conditions. Les possibilités d'améliorations en ce sens étant néanmoins limitées, nous élargissons la perspective en considérant des réseaux de Chebyshev avec singularités. Notre résultat principal est l'existence de réseaux de Chebyshev avec singularités coniques, lisses par morceaux, sur toute surface dont la courbure totale positive est inférieure à $2pi$ et dont la courbure totale négative est finie. Notre preuve est constructive, ce qui permet de déterminer ces réseaux dans des cas pratiques. Nous avons implémenté un cas particulier de notre algorithme dans le logiciel Rhinoceros et nous présentons des exemples de réseaux construits par cette méthode / Chebyshev nets are coordinate systems on surfaces obtained by pure shearing of a planar domain.These nets are used in particular to model gridshells, an architectural construction which is well-known for its low environmental impact. The main issue when designing a gridshell is the lack of diversityof the accessible shapes. Indeed, although any surface admits locally a Chebyshev net at any point, the global existence for these coordinate systems is only possible for a restricted set of surfaces. The research for sufficient conditions ensuring the global existence of Chebyshev nets is still ongoing. A result achieved in this thesis is an improvement on these conditions. Since the improvement in this direction seems to be rather limited, we broaden the perspective by introducing Chebyshev nets with singularities. Our main result is the existence of a global Chebyshev net with conical singularities on any surface with total positive curvature less than $2pi$ and with finite total negative curvature. Our proof is constructive, so that this method can be applied to practical cases. We have implemented a special instance of this algorithm in the software Rhinoceros and some discrete Chebyshev nets constructed using this method are presented

This project presents a modified method of numerical integration for a “well behaved� function over the finite interval [-1,1]. Similar to the Clenshaw-Curtis quadrature rule, this new algorithm relies on expressing the integrand as an expansion of Chebyshev polynomials of the second kind. The truncated series is integrated term-by-term yielding an approximation for the integral of which we wish to compute. The modified method is then contrasted with its predecessor Clenshaw-Curtis, as well as the classical method of Gauss-Legendre in terms of convergence behavior, error analysis and computational efficiency. Lastly, illustrative examples are shown which demonstrate the dependence that the convergence has on the given function to be integrated.

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