  We study multivariate $\boldsymbol{L}_{\infty}$-approximation for a weighted
Korobov space of periodic functions for which the Fourier coefficients decay
exponentially fast. The weights are defined, in particular, in terms of two
sequences $\boldsymbol{a}=\{a_j\}$ and $\boldsymbol{b}=\{b_j\}$ of positive
real numbers bounded away from zero. We study the minimal worst-case error
$e^{\boldsymbol{L}_{\infty}\mathrm{-app},\Lambda}(n,s)$ of all algorithms that
use $n$ information evaluations from a class $\Lambda$ in the $s$-variate case.
We consider two classes $\Lambda$ in this paper: the class $\Lambda^{{\rm
all}}$ of all linear functionals and the class $\Lambda^{{\rm std}}$ of only
function evaluations.
  We study exponential convergence of the minimal worst-case error, which means
that $e^{\boldsymbol{L}_{\infty}\mathrm{-app},\Lambda}(n,s)$ converges to zero
exponentially fast with increasing $n$. Furthermore, we consider how the error
depends on the dimension $s$. To this end, we define the notions of
$\kappa$-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC
stands for "exponential convergence". In particular, EC-polynomial tractability
means that we need a polynomial number of information evaluations in $s$ and
$1+\log\,\varepsilon^{-1}$ to compute an $\varepsilon$-approximation. We derive
necessary and sufficient conditions on the sequences $\boldsymbol{a}$ and
$\boldsymbol{b}$ for obtaining exponential error convergence, and also for
obtaining the various notions of tractability. The results are the same for
both classes $\Lambda$.
