A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon

Abstract

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.