  Following Frantzikinakis' approach on averages for Hardy field functions of
different growth, we add to the topic by studying the corresponding averages
for tempered functions, a class which also contains functions that oscillate
and is in general more restrictive to deal with. Our main result is the
existence and the explicit expression of the $L^2$-norm limit of the
aforementioned averages, which turns out, as in the Hardy field case, to be the
"expected" one. The main ingredients are the use of, the now classical, PET
induction (introduced by Bergelson), covering a more general case, namely a
"nice" class of tempered functions (developed by Chu-Frantzikinakis-Host for
polynomials and Frantzikinakis for Hardy field functions) and some
equidistribution results on nilmanifolds (analogous to the ones of
Frantzikinakis' for the Hardy field case).
