We study the recovery of functions in the uniform norm based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best $L_2$-approximation from a given nested sequence of subspaces combined with bounds on the the Christoffel function of these subspaces. Besides an explicit bound, we obtain that linear algorithms using $n$ samples are optimal up to a factor $\sqrt{n}$ among all algorithms using arbitrary linear information. Moreover, our results imply that linear sampling algorithms are optimal up to a constant factor for many reproducing kernel Hilbert spaces. We also discuss results for approximation in more general seminorms, including $L_p$-approximation.

翻译：我们研究基于函数评估的均匀范数下的函数恢复问题。我们基于给定嵌套子空间列的最佳L₂逼近误差及这些子空间的克里斯托费尔函数边界，获得了函数类在最坏情况下的误差上界。除显式上界外，我们证明：在使用n个样本的线性算法中，其性能与任意使用线性信息的算法相比，至多相差因子√n。此外，我们的结果表明：对许多再生核希尔伯特空间而言，线性采样算法与最优算法相比仅差常数因子。我们还讨论了更一般半范数（包括Lp逼近）下的近似结果。