Trigonometric Interpolation and Quadrature in Perturbed Points

TL;DR

This paper investigates the convergence of trigonometric interpolants and quadrature formulas when the interpolation points are perturbed, establishing conditions under which convergence is guaranteed based on the function's smoothness and the perturbation magnitude.

Contribution

It extends classical results by showing convergence for perturbed equispaced points when the function is sufficiently smooth, with bounds related to the perturbation size.

Findings

Convergence holds for all perturbations with <1/2 if the function has 4 derivatives.

The convergence rate depends on the smoothness of the function.

Conjecture that 2 derivatives may suffice for convergence.

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.