On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

Abstract

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the $L_{\infty }$ norm. We consider algorithms that use standard information ${\Lambda }^{\mathrm{std}}$ consisting of function values or general linear information ${\Lambda }^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for ${\Lambda }^{\mathrm{std}}$ and ${\Lambda }^{\mathrm{all}}$ under the absolute or normalized error criterion, and show that the power of ${\Lambda }^{\mathrm{std}}$ is the same as the one of ${\Lambda }^{\mathrm{all}}$ for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].