  When a function $f(x)$ is singular at a point $x_{s}$ on the real axis, its
Fourier series, when truncated at the $N$-th term, gives a pointwise error of
only $O(1/N)$ over the entire real axis. Such singularities spontaneously arise
as "fronts" in meteorology and oceanography and "shocks" in other branches of
fluid mechanics. It has been previously shown that it is possible to recover an
exponential rate of convegence at all points away from the singularity in the
sense that $|f(x) - f_{N}^{\sigma}(x) | \sim O(\exp(- q(x) N))$ where
$f_{N}^{\sigma}(x)$ is the result of applying a filter or summability method to
the partial sum $f_{N}(x)$ and $q(x)$ is a proportionality constant that is a
function of $d(x) \equiv |x-x_{s}|$, the distance from $x$ to the singularity.
Here we give an elementary proof of great generality using conformal mapping in
a dummy variable $z$; this is equivalent to applying the Euler acceleration. We
show that $q(x) \approx \log(\cos(d(x)/2))$ for the Euler filter when the
Fourier period is $2 \pi$. More sophisticated filters can increase $q(x)$, but
the Euler filter is simplest. We can also correct recently published claims
that only a root-exponential rate of convergence can be recovered for filters
of compact support such as the Euler acceleration and the Erfc-Log filter.
