  The problem of tolerant junta testing is a natural and challenging problem
which asks if the property of a function having some specified correlation with
a $k$-Junta is testable. In this paper we give an affirmative answer to this
question: We show that given distance parameters $\frac{1}{2} >c_u>c_{\ell} \ge
0$, there is a tester which given oracle access to $f:\{-1,1\}^n \rightarrow
\{-1,1\}$, with query complexity $ 2^k \cdot \mathsf{poly}(k,1/|c_u-c_{\ell}|)$
and distinguishes between the following cases:
  $\mathbf{1.}$ The distance of $f$ from any $k$-junta is at least $c_u$;
  $\mathbf{2.}$ There is a $k$-junta $g$ which has distance at most $c_\ell$
from $f$.
  This is the first non-trivial tester (i.e., query complexity is independent
of $n$) which works for all $1/2 > c_u > c_\ell \ge 0$. The best previously
known results by Blais \emph{et~ al.}, required $c_u \ge 16 c_\ell$. In fact,
with the same query complexity, we accomplish the stronger goal of identifying
the most correlated $k$-junta, up to permutations of the coordinates.
  We can further improve the query complexity to $\mathsf{poly}(k,
1/|c_u-c_{\ell}|)$ for the (weaker) task of distinguishing between the
following cases:
  $\mathbf{1.}$ The distance of $f$ from any $k'$-junta is at least $c_u$.
  $\mathbf{2.}$ There is a $k$-junta $g$ which is at a distance at most
$c_\ell$ from $f$.
  Here $k'=O(k^2/|c_u-c_\ell|)$.
  Our main tools are Fourier analysis based algorithms that simulate oracle
access to influential coordinates of functions.
