On the power of standard information for tractability for $L_{\mathrm{\infty }}$ approximation of periodic functions in the worst case setting

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