Trigonometric Interpolation and Quadrature in Perturbed Points

Author: 

Austin, AP
Trefethen, LN

Publication Date: 

6 September 2017

Journal: 

SIAM Journal on Numerical Analysis

Last Updated: 

2018-10-13T03:00:05.01+01:00

abstract: 

The trigonometric interpolants to a periodic function $f$ in equispaced
points converge if $f$ is Dini-continuous, and the associated quadrature
formula, the trapezoidal rule, converges if $f$ is continuous. What if the
points are perturbed? With equispaced grid spacing $h$, let each point be
perturbed by an arbitrary amount $\le \alpha h$, where $\alpha\in [\kern .5pt
0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests
there may be be trouble for $\alpha\ge 1/4$. We show that convergence of both
the interpolants and the quadrature estimates is guaranteed for all
$\alpha<1/2$ if $f$ is twice continuously differentiable, with the convergence
rate depending on the smoothness of $f$. More precisely it is enough for $f$ to
have $4\alpha$ derivatives in a certain sense, and we conjecture that $2\alpha$
derivatives is enough. Connections with the Fej\'er--Kalm\'ar theorem are
discussed.

Symplectic id: 

666274

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Submitted to ORA: 

Submitted

Publication Type: 

Journal Article