  We say that a family $\mathcal F$ of $k$-element sets is a {\it $j$-junta} if
there is a set $J$ of size $j$ such that, for any $F$, its presence in
$\mathcal F$ depends on its intersection with $J$ only. Approximating arbitrary
families by $j$-juntas with small $j$ is a recent powerful technique in
extremal set theory.
  The weak point of all known junta approximation results is that they work in
the range $n>Ck$, where $C$ is an extremely fast growing function of the input
parameters, such as the quality of approximation or the number of families we
simultaneously approximate.
  We say that a family $\mathcal F$ is {\it shifted} if for any
$F=\{x_1,\ldots, x_k\}\in \mathcal F$ and any $G =\{y_1,\ldots, y_k\}$ such
that $y_i\le x_i$, we have $G\in \mathcal F$. For many extremal set theory
problems, including the Erd\H os Matching Conjecture, or the Complete
$t$-Intersection Theorem, it is sufficient to deal with shifted families only.
  In this paper, we present very general approximation by juntas results for
shifted families with explicit (and essentially linear) dependency on the input
parameters. The results are best possible up to some constant factors.
Moreover, they give meaningful statements for almost all range of values of
$n$. The proofs are shorter than the proofs of the previous approximation by
juntas results and are completely self-contained.
  As an application of our junta approximation, we give a nearly-linear bound
for the multi-family version of the Erd\H os Matching Conjecture. More
precisely, we prove the following result. Let $n\ge 12sk\log(e^2s)$ and suppose
that the families $\mathcal F_1,\ldots, \mathcal F_s\subset {[n]\choose k}$ do
not contain $F_1\in\mathcal F_1,\ldots, F_s\in \mathcal F_s$ such that $F_i$'s
are pairwise disjoint. Then $\min_{i}|\mathcal F_i|\le {n\choose
k}-{n-s+1\choose k}.$
