Sparse-grid sampling recovery and numerical integration of functions having mixed smoothness (Q6650705)

This paper provides a survey of recent advancements in sparse-grid linear algorithms for the approximate recovery and integration of functions with mixed smoothness. The author presents a collection of results that extend the understanding of sampling recovery and numerical integration for functions defined on a compact subset in \( \mathbb{R}^d \) or the entire space \( \mathbb{R}^d \). The paper is structured to cover both unweighted and weighted settings, with a focus on the optimality of these methods in terms of linear sampling widths and numerical integration errors.