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, we analyze how the uneven distribution of the samples impacts the quality of interpolation and quadrature. Starting with equally spaced nodes on [-π,π) with grid spacing h, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount ≤α h, where 0 ≤α < 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 α < 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 ≤α < 1/4. Though, quadrature rules at perturbed nodes can have negative weights for any α > 0, we provide a bound on the absolute sum of the quadrature weights. Therefore, we show that perturbed equally spaced grids with small α can be used without numerical woes. While our proof techniques work primarily when 0 ≤α < 1/4, we show that a small amount of oversampling extends our results to the case when 1/4 ≤α < 1/2.
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