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.

翻译：通过比较Gegenbauer预测和最佳近效的趋同行为,我们研究了Gegenbauer预测在最大规范中的最佳趋同率。我们表明,Gegenbauer预测的趋同率与在基本功能条件下的最佳近比率相同。我们显示,Gegenbauer预测的趋同率与在基本功能条件下的最佳近似率相同,要么对椭圆和美元(lambdaleq)和美元(lambda\leq$或差异)进行分析,要么在椭圆和美元(lambda$)或美元(lambda$是Gegenba美元)和美元(lambda美元)和美元(美元)的相异和美元(lambda\leq美元)的趋同率,在Gegenba预测的趋同率最佳趋同率和欧元区域预测的最佳趋同率相同。 与Gegenba预测相比,内部或Gebarbaririmal 的精确性预测的功能更慢于最佳的理论解释。