  On a smooth algebraic variety over $\mathbb{C}$, we build the tempered
subanalytic and Stein tempered subanalytic sites. We construct the sheaf of
holomorphic functions tempered at infinity over these sites and study their
relations with the sheaf of regular functions, proving in particular that these
sheaves are isomorphic on Zariski open subsets. We show that these data allow
to define the functors of tempered and Stein tempered analytifications. We
study the relations between these two functors and the usual analytification
functor. We also obtain algebraization results in the non-proper case and
flatness results.
