Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

Abstract

Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for $L_{\infty }$-approximation, with precise control of all involved constants. As an illustration, we derive some results for periodic isotropic Sobolev spaces $H^s({\mathbb{T}}^d)$ and Sobolev spaces of dominating mixed smoothness $H_{\mathrm{mix}}^s({\mathbb{T}}^d)$, always equipped with natural norms. Some results for isotropic as well as dominating mixed Besov spaces are also obtained.