부산대학교 도서관

Orthogonal polynomials are frequently related to different fields of research, such as approximation theory or numerical analysis, and play a crucial role in numerous applications, such as the construction of Gaussian quadrature formulas, the resolution of the Gibbs phenomenon or spectral methods for the numerical solutions of differential equations. \par Due to the important role played by orthogonal polynomials in various areas of mathematics and physics, their approximation properties have attracted considerable interest, especially in the spectral methods community. \par In this paper, the author addresses the study of these approximation properties in the specific case of Gegenbauer polynomials. More precisely, he compares the convergence behavior of Gegenbauer projections and best approximations by analyzing the optimal convergence rates of Gegenbauer projections in the maximum norm. In the case of analytic functions, he establishes some explicit error bounds for Gegenbauer projections, in the maximum norm, and shows that these bounds are optimal in the sense that they cannot be improved. In the case of piecewise analytic functions of class $C^{m-1}$ with ${m\in \Bbb{N}}$, the author establishes optimal convergence rates of Gegenbauer projections in the maximum norm. He also studies optimal convergence rates of Gegenbauer projections for functions with algebraic singularities. \par The author also provides an explanation for the error localization property of Gegenbauer projections and shows that Gegenbauer projections are actually more accurate than the best approximations, except in the neighborhood of critical points. \par All these results are illustrated by numerical experiments.