We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ 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 $\Lambda^{\rm all}$ for the absolute error criterion, and show that the power of $\Lambda^{\rm std}$ is the same as that of $\Lambda^{\rm all}$ for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by E.Novak and H.Wo\'zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Z\"urich, 2012.

翻译：我们研究平均案例设置中的多变量近似值, 其误差以加权 $2 标准值计算。 我们考虑使用标准信息 $Lambda\\rm std}$\Lambda\rm std}$\Lamda\rm all}$Lambda\rm all}$的算法, 由任意连续线性函数构成的函数值或一般线性信息 $Lambda\rm all}$Lambda\Lamda\rm all}$。 我们调查各种变数概念的等值值和指数性可移动性, $Lambda\rm std}$。 我们调查各种变数和指数性可移动性概念的等值, $Lambda\\rm\rm\ all} $。 我们具体地调查“ E. Novak 和 H. Wo\\' zniakowski” 的书中的“ 多变数问题的可性, 卷 第三卷: 标准信息, III: 操作者的标准信息, 数学中的 EMS 类, 2012”