부산대학교 도서관

Orthogonal polynomials are frequently related to different fields of research, such as approximation theory or numerical analysis, and play a crucialrole in numerous applications, such as the construction of Gaussianquadrature formulas, the resolution of the Gibbs phenomenon or spectralmethods for the numerical solutions of differential equations.\par Due to the important role played by orthogonal polynomials in variousareas of mathematics and physics, their approximation properties haveattracted considerable interest, especially in thespectral methods community.\par In this paper, the author addresses the study of these approximationproperties in the specific case of Gegenbauer polynomials. Moreprecisely, he compares the convergence behavior of Gegenbauerprojections and best approximations by analyzing the optimalconvergence rates of Gegenbauer projections in the maximum norm. In thecase of analytic functions, he establishes some explicit error boundsfor Gegenbauer projections, in the maximum norm, and shows that thesebounds are optimal in the sense that they cannot be improved. In thecase of piecewise analytic functions of class $C^{m-1}$ with ${m\in\Bbb{N}}$, the author establishes optimal convergence rates of Gegenbauerprojections in the maximum norm. He also studies optimal convergencerates of Gegenbauer projections for functions with algebraicsingularities.\par The author also provides an explanation for the error localizationproperty of Gegenbauer projections and shows that Gegenbauerprojections are actually more accurate than the best approximations,except in the neighborhood of critical points.\par All these results are illustrated by numerical experiments.