Worst case tractability of $L_2$-approximation for weighted Korobov spaces

We study $L_2$-approximation problems ${\text{APP}}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\text{a},}$ with parameter sequences $=\{{\text{ga}}_j\}$ and $\text{a}=\{{\text{az}}_j\}$ of positive real numbers $1\ge {\text{ga}}_1\ge {\text{ga}}_2\ge \cdots \ge 0$ and $\frac{1}{2}<{\text{az}}_1\le {\text{az}}_2\le \cdots$. 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 ${\text{va}}^{-1}$ with the accuracy $\text{va}$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences $$ and $\text{a}$ for strongly polynomial tractability or polynomial tractability is $$\text{dz}:=\underset{j\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\ln {\text{ga}}_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max \{\frac{1}{\text{dz}},\frac{1}{2{\text{az}}_1}\}\mathrm{.}$$