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 Nth term, gives a pointwise error of only O(1∕N) over the entire real axis. Boyd and Moore [Summability methods for Hermite functions. Dyn. Atmos. Oceans 10, 51–62 (1986)] and Boyd [A lag-averaged generalization of Euler’s method for accelerating series. Appl. Math. Comput. 72, 146–166 (1995)] proved that it is possible to recover an exponential rate of convegence at all points away from the singularity in the sense that $\vert f(x) - {f}_{N}^{\sigma }(x)\vert \sim O(\exp (-\mu (x)N))$ where ${f}_{N}^{\sigma }(x)$ is the result of applying a summability method to the partial sum f _N (x) and μ(x) is a proportionality constant that is a function of $d(x) \equiv \vert x - {x}_{s}\vert$, the distance from x to the singularity. Here we improve these earlier results and give an elementary proof of great generality using conformal mapping in a dummy variable z, which is the Euler acceleration. We show $\exp (\mu (x)) \approx \min (2,\cos (d(x)/2))$ for the Euler filter when the Fourier period is 2π and f(x) has no off-axis singularities very close to the real axis. We correct recent claims that only a root-exponential rate of convergence can be recovered.