Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 117-118).

Esta tese analisa o método de colocação polinomial, para uma classe de equações integro-diferenciais singulares em espaços ponderados de funções contínuas e condições de fronteira não nulas. A convergência do método numérico em espaços com norma uniforme ponderada, é demonstrada, e taxas de convergências são determinadas, usando a suavidade dos dados das funções envolvidas no problema. Exemplos numéricos confirmam as estimativas / This thesis analyses the polynomial collocation method, for a class of singular integro-differential equations in weighted spaces of continuous functions, and non-homogeneous boundary conditions. Convergence of the numerical method, in weighted uniform norm spaces, is demonstrated and convergence rates are determined using the smoothness of the data functions involved in problem. Numerical examples confirm the estimates

The Besov class $B_{pq}^s$ is defined by ${ f : { 2^{|n|s}||W_n*f||_p } _{ninmathbb{Z}}in ell^q(mathbb{Z}) }$. When $s=1$, $p=q $, we know if $f in B_p$ if and only if $int_mathbb{D} |f^{(n)}(z)|^p(1-|z|^2)^{2pn-2}dv(z) <+infty$. It is shown in [5] that $int_{mathbb{D}}|f^{'}(z)|^q K(z,z)^{1-q}dv(z)= O(L(b(e^{-(q-p)^{-1}})))$ if $f in B_{L,p}$. In this paper we will show that $f in B_{L,p}$ if and only if $sum_{n=0}^{infty}2^{nq}||W_n*f||_p^q = O(L(b(e^{-(q-p)^{-1}})))$.

Thèse de doctorat : Mathématiques : Paris 12 : 2005. / Titre provenant de l'écran-titre. Bibliogr. : 99 réf. Index.

Thesis (Ph. D.)--University of Missouri-Columbia, 2008. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 18, 2009) Vita. Includes bibliographical references.

Esta tese analisa o método de colocação polinomial, para uma classe de equações integro-diferenciais singulares em espaços ponderados de funções contínuas e condições de fronteira não nulas. A convergência do método numérico em espaços com norma uniforme ponderada, é demonstrada, e taxas de convergências são determinadas, usando a suavidade dos dados das funções envolvidas no problema. Exemplos numéricos confirmam as estimativas / This thesis analyses the polynomial collocation method, for a class of singular integro-differential equations in weighted spaces of continuous functions, and non-homogeneous boundary conditions. Convergence of the numerical method, in weighted uniform norm spaces, is demonstrated and convergence rates are determined using the smoothness of the data functions involved in problem. Numerical examples confirm the estimates

We look at the mixed boundary value problem for elliptic operators in a bounded C1,1(ℝn) domain. The boundary is decomposed into disjoint parts, D and N, with Dirichlet and Neumann data, respectively. Expanding on work done by Ott and Brown, we find a larger range of values of p, 1 < p < n/(n-1), for which the Lp mixed problem has a unique solution with the non-tangential maximal function of the gradient in Lp(∂Ω).

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 27, 2007) Vita. Includes bibliographical references.

This thesis consists of an introduction and a summary, followed by two papers, both of them on the topic of function spaces on fractals. Paper I: Andreas Brodin, Pointwise Convergence of Haar type Wavelets on Self-Similar Sets, Manuscript. Paper II: Andreas Brodin, Regularization of Wavelet Expansion characterizes Besov Spaces on Fractals, Manuscript. Properties of wavelets, originally constructed by A. Jonsson, are studied in both papers. The wavelets are piecewise polynomial functions on self-similar fractal sets. In Paper I, pointwise convergence of partial sums of the wavelet expansion is investigated. On a specific fractal set, the Sierpinski gasket, pointwise convergence of the partial sums is shown by calculating the kernel explicitly, when the wavelets are piecewise constant functions. For more general self-similar fractals, pointwise convergence of the partial sums and their derivatives, in case the expanded function has higher regularity, is shown using a different technique based on Markov's inequality. A. Jonsson has shown that on a class of totally disconnected self-similar sets it is possible to characterize Besov spaces by means of the magnitude of the coefficients in the wavelet expansion of a function. M. Bodin has extended his results to a class of graph directed self-similar sets introduced by Mauldin and Williams. Unfortunately, these results only holds for fractals such that the sets in the first generation of the fractal are disjoint. In Paper II we are able to characterize Besov spaces on a class of fractals not necessarily sharing this condition by making the wavelet expansion smooth. We create continuous regularizations of the partial sums of the wavelet expansion and show that properties of these regularizations can be used to characterize Besov spaces.

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