We study the recovery of functions in various norms, including $L_p$ with $1\le p\le\infty$, 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 and the Christoffel function of these subspaces. In the case $p=\infty$, our results imply that linear sampling algorithms are optimal up to a constant factor for many reproducing kernel Hilbert spaces.

翻译：我们研究了基于函数求值、在多种范数（包括 $1\le p\le\infty$ 时的 $L_p$ 范数）下函数的恢复问题。我们针对一般的函数类，以给定嵌套子空间序列的最佳 $L_2$ 逼近以及这些子空间的 Christoffel 函数为度量，获得了最坏情况下的误差界。在 $p=\infty$ 的情形下，我们的结果表明，对于许多再生核希尔伯特空间，线性采样算法在常数因子范围内是最优的。