On the stability of unevenly spaced samples for interpolation and quadrature

TL;DR

This paper investigates how small perturbations in evenly spaced samples affect interpolation and quadrature, proving stability results and bounds for perturbations up to a certain threshold.

Contribution

It establishes a discrete Kadec-1/4 theorem for perturbed nodes and analyzes the convergence and stability of interpolation and quadrature under these perturbations.

Findings

Bounded condition number for nonuniform discrete Fourier transform when perturbation < 1/4

Quadrature rules converge for continuous functions under small perturbations

Quadrature weights can be negative but have bounded absolute sum

Abstract

Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, we analyze how the uneven distribution of the samples impacts the quality of interpolation and quadrature. Starting with equally spaced nodes on $[-\pi,\pi)$ with grid spacing $h$, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount $\leq \alpha h$, where $0 \leq \alpha < 1/2$ is a fixed constant. We prove a discrete version of the Kadec-1/4 theorem, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of $h$, for any $\alpha < 1/4$. We go on to show that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when $0 \leq \alpha < 1/4$. Though, quadrature rules at perturbed nodes can have negative weights for any $\alpha > 0$, we provide a bound on the absolute sum of the quadrature weights. Therefore, we show that perturbed equally spaced grids with small $\alpha$ can be used without numerical woes. While our proof techniques work primarily when $0 \leq \alpha < 1/4$, we show that a small amount of oversampling extends our results to the case when $1/4 \leq \alpha < 1/2$.