  Let $Z$ be a unimodular real spherical space. We develop a theory of constant
terms for tempered functions on $Z$ which parallels the work of Harish-Chandra.
The constant terms $f_I$ of an eigenfunction $f$ are parametrized by subsets
$I$ of the set $S$ of spherical roots which determine the fine geometry of $Z$
at infinity. Constant terms are transitive i.e. $(f_J)_I=f_I$ for $I\subset J$,
and our main result is a quantitative bound of the difference $f-f_I$, which is
uniform in the parameter of the eigenfunction.
