Optimal rates of convergence and error localization of Gegenbauer projections

TL;DR

This paper analyzes the convergence rates of Gegenbauer projections, showing they match best approximations under certain conditions and explaining their error localization behavior, which enhances understanding of spectral methods.

Contribution

It provides the first detailed analysis of the optimal convergence rates and error localization of Gegenbauer projections, revealing conditions where they outperform or match best approximations.

Findings

Gegenbauer projections converge at the same rate as best approximations for analytic functions with $ ext{Re}( ext{elliptic domain})$ and $ ext{Re}( ext{differentiability})$ conditions.

For analytic functions with $ ext{Re}( ext{elliptic domain})$ and $ ext{Re}( ext{differentiability})$, convergence is slower by factors of $n^{ ext{parameter}}$.

Error localization explains improved accuracy near critical points for functions with singularities.

Abstract

Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and $\lambda\leq0$ or differentiable and $\lambda\leq1$, where $\lambda$ is the parameter in Gegenbauer projections. If the underlying function is analytic and $\lambda>0$ or differentiable and $\lambda>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^{\lambda}$ and $n^{\lambda-1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $\lambda>-1/2$. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods.