We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that use $n$ arbitrary continuous linear functionals with $d$ variables. We study polynomial convergence of the minimal worst case error, which means that $e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We recall the notions of polynomial, strongly polynomial, weak and $(t_1,t_2)$-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in $d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$

翻译：我们研究在加权Korobov空间$H_{d,\a,{\bm \ga}}$中，对于参数序列${\bm \ga}=\{\ga_j\}$和$\a=\{\az_j\}$（其中正实数满足$1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$且$\frac1 2<\az_1\le \az_2\le \cdots$），最坏情形下$L_2$逼近问题$\text{APP}_d$。考虑使用$d$个变量的$n$个任意连续线性泛函的算法的最小最坏情形误差$e(n,\text{APP}_d)$。我们研究最小最坏情形误差的多项式收敛性，即随着$n$增大，$e(n,\text{APP}_d)$以多项式速度收敛到零。回顾多项式可处理性、强多项式可处理性、弱可处理性和$(t_1,t_2)$-弱可处理性的概念。特别地，多项式可处理性意味着在逼近精度$\va$下，需要关于$d$和$\va^{-1}$的多项式数量的任意连续线性泛函。我们得到，对于强多项式可处理性或多项式可处理性，序列${\bm \ga}$和$\a$的匹配充要条件为：$$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$且强多项式可处理性的指数为：$$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$