English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

When a function $f(x)$ is singular at a point $x_{s}$ on the real axis, itsFourier series, when truncated at the $N$-th term, gives a pointwise error ofonly $O(1/N)$ over the entire real axis. Such singularities spontaneously ariseas "fronts" in meteorology and oceanography and "shocks" in other branches offluid mechanics. It has been previously shown that it is possible to recover anexponential rate of convegence at all points away from the singularity in thesense 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 tothe partial sum $f_{N}(x)$ and $q(x)$ is a proportionality constant that is afunction 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 ina dummy variable $z$; this is equivalent to applying the Euler acceleration. Weshow that $q(x) \approx \log(\cos(d(x)/2))$ for the Euler filter when theFourier period is $2 \pi$. More sophisticated filters can increase $q(x)$, butthe Euler filter is simplest. We can also correct recently published claimsthat only a root-exponential rate of convergence can be recovered for filtersof compact support such as the Euler acceleration and the Erfc-Log filter.

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