关于插值与求积的非均匀分布样本的稳定性
On the stability of unevenly spaced samples for interpolation and quadrature
AI总结：
本文分析了周期函数非均匀分布样本对插值与求积质量的影响，证明了Kadec-1/4定理的离散版本，并指出小扰动均匀网格可无数值隐患使用，少量过采样可扩展结果适用范围。
摘要：
来自周期函数的非均匀分布样本在信号处理中很常见，通常可被视为扰动的均匀网格。本文分析了样本的非均匀分布如何影响插值和求积的质量。从 $[-π,π)$ 上网格间距为 $h$ 的均匀节点出发，假设非均匀节点是通过将每个均匀节点扰动任意量 $\leq αh$ 获得的，其中 $0 \leq α< 1/2$ 是固定常数。我们证明了 Kadec-1/4 定理的离散版本，指出对于任何 $α< 1/4$，与扰动节点相关的非均匀离散傅里叶变换具有独立于 $h$ 的有界条件数。我们进一步证明，当 $0 \leq α< 1/4$ 时，非均匀求积规则对所有连续函数收敛，且插值对所有导数具有有界变差的可微函数一致收敛。尽管扰动节点的求积规则在任意 $α> 0$ 时可能具有负权重，我们仍给出了求积权重绝对和的界限。因此，小 $α$ 的扰动均匀网格可无数值隐患地使用。虽然证明技术主要适用于 $0 \leq α< 1/4$，但我们表明少量过采样可将结果扩展至 $1/4 \leq α< 1/2$ 的情况。
英文摘要：
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 $\leq αh$, where $0 \leq α< 1/2$ 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 $0 \leq α< 1/4$. 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 $0 \leq α< 1/4$, we show that a small amount of oversampling extends our results to the case when $1/4 \leq α< 1/2$.
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