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, also in $L_p$-norm, 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. Our results imply that linear sampling algorithms are optimal (up to constants) among all algorithms using arbitrary linear information for many reproducing kernel Hilbert spaces; a result that has been observed independently in [Geng \& Wang, arXiv:2304.14748].

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。对于一般函数类，我们给出最坏情况误差界（同样适用于$L_p$-范数），该误差界由给定嵌套子空间序列的最优$L_2$-逼近误差，并结合这些子空间的Christoffel函数界共同决定。我们的结果表明：对许多再生核希尔伯特空间而言，在任意线性信息的所有算法中，线性采样算法是最优的（至多相差常数因子）。这一结论在[Geng \& Wang, arXiv:2304.14748]中已被独立观察到。