论文标题
Gegenbauer预测的收敛性和错误定位率
Optimal rates of convergence and error localization of Gegenbauer projections
论文作者
论文摘要
通过比较Gegenbauer预测和最佳近似值的收敛行为的动机，我们研究了Gegenbauer投影的最佳收敛速率，以最大的规范。我们表明，Gegenbauer预测的收敛速率与基础功能条件下的最佳近似值相同，要么是在椭圆形和$λ\ leq0 $或可区分和$λ\ leq1 $的分析中，其中$λ$是Gegenbauer Projictions的参数。如果基本功能是分析性的，$λ> 0 $或可区分的$λ> 1 $，则分别按$ n^λ$和$ n^λ$和$ n^{λ-1-1} $的因素的gegenbauer预测收敛速度要慢。一个特殊情况是具有端点奇点的功能，对于所有$λ> -1/2 $，Gegenbauer的预测和最佳近似值都以相同的速率收敛。对于具有内部或端点奇点的功能，我们为Gegenbauer预测的错误定位现象提供了理论上的解释，以及为什么Gegenbauer预测的准确性比在关键点的小社区中要好于最佳近似值。我们的分析从根本上为Gegenbauer近似和相关光谱方法的力量提供了新的见解。
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 $λ\leq0$ or differentiable and $λ\leq1$, where $λ$ is the parameter in Gegenbauer projections. If the underlying function is analytic and $λ>0$ or differentiable and $λ>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^λ$ and $n^{λ-1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $λ>-1/2$. 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.
