We study multivariate approximation of periodic function in the worst case setting with the error measured in the $L_\infty$ norm. We consider algorithms that use standard information $\Lambda^{\rm std}$ consisting of function values or general linear information $\Lambda^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $\Lambda^{\rm std}$ and $\Lambda^{\rm all}$ under the absolute or normalized error criterion, and show that the power of $\Lambda^{\rm std}$ is the same as the one of $\Lambda^{\rm all}$ 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 Wo\'zniakowski (2012).

翻译：我们研究了最坏情况下周期函数的多变量逼近问题，其中误差采用$L_\infty$范数度量。我们考虑使用标准信息$\Lambda^{\rm std}$（由函数值构成）或一般线性信息$\Lambda^{\rm all}$（由任意连续线性泛函构成）的算法。在绝对或归一化误差准则下，我们研究了$\Lambda^{\rm std}$与$\Lambda^{\rm all}$在各类代数及指数可处理性概念上的等价性，并证明了对于部分代数与指数可处理性概念，$\Lambda^{\rm std}$的能力与$\Lambda^{\rm all}$相同。该结果可应用于带权重的Korobov空间及指数权重的Korobov空间，从而为Novak和Woźniakowski (2012)提出的开放问题145提供了特定解。