Worst case tractability of L2L_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,\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\}.$