We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were obtained only recently. We discuss different aspects of the information-theoretic limit that appears because of the limited amount of data available, as well as algorithms and sampling strategies that come as close to it as possible. We also discuss (optimal) sampling in a broader sense, allowing other types of measurements that may be nonlinear, adaptive and random, and present several relations between the different settings in the spirit of information-based complexity. We hope that this article provides both, a basic introduction to the subject and a contemporary summary of the current state of research.

翻译：我们考虑基于有限个函数评估的函数逼近或恢复问题。这是最优恢复、机器学习以及数值分析领域中一个广泛研究的问题，但许多基本见解直到近期才被获得。我们探讨了因可用数据有限而导致的信息论极限的不同方面，以及尽可能接近该极限的算法与采样策略。我们还从更广义的角度讨论了（最优）采样，允许其他类型的测量方式，这些方式可以是非线性的、自适应的且随机的，并呈现了基于信息复杂性思想的不同设定之间的若干关系。我们希望本文既能作为该主题的基础入门，也能提供当前研究现状的当代综述。