On the stability of unevenly spaced samples for interpolation and quadrature

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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, the question of how the uneven distribution of the samples impacts the quality of interpolation and quadrature is analyzed. 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 $$\le \alpha h$$ , where $$0 \le \alpha < 1/2$$ is a fixed constant. A discrete version of the Kadec-1/4 theorem is proved, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of h, for any $$\alpha <1/4$$ . Then, it is shown that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when $$0\le \alpha <1/4$$ . Though, quadrature rules at perturbed nodes can have negative weights for any $$\alpha >0$$ , a bound on the absolute sum of the quadrature weights is provided, which shows that perturbed equally spaced grids with small $$\alpha $$ can be used without numerical woes. While the proof techniques work primarily when $$0 \le \alpha < 1/4$$ , it is shown that a small amount of oversampling extends our results to the case when $$1/4\le \alpha <1/2$$ .