We present an asymptotic analysis of adaptive methods for L^p approximation of functions f ∈ C^r([a, b]), where \(1\le p\le +\infty \). The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m^−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic L^p approximation with a given accuracy ε.

中文翻译：

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使用渐近最优自适应插值的自动逼近

我们提出的自适应方法的一个渐近分析大号^p的函数逼近˚F ∈ Ç ^{- [R}（[一，b ]），其中\（1个\文件p \文件+ \ infty \） 。这些方法依赖于度数r -1的分段多项式插值以及选择m个子间隔的自适应策略。在这种情况下，最佳收敛速度为m ^{− r}并且已经通过初始间隔的统一（非自适应）细分来实现；然而，渐近常数关键取决于所选择的策略。我们得出了渐近最佳的自适应策略，并显示了它们在给定精度ε下对自动L ^p逼近的适用性。