论文标题
关于用于插值和正交的不均间隔样品的稳定性
On the stability of unevenly spaced samples for interpolation and quadrature
论文作者
论文摘要
来自周期函数的不均间隔在信号处理中很常见，通常可以看作是扰动的同等间隔网格。在本文中，我们分析了样品的不均匀分布如何影响插值和正交的质量。从$ [ - π，π）上的等距间隔节点开始，并带有网格间距$ h $，假设通过将每个统一节点以任意量$ \ leqleqαH$扰动而获得的不均匀间隔节点，其中$ 0 \ leq leqleqα<1/2 $是固定常数。我们证明了KADEC-1/4定理的离散版本，该版本指出，与任何$α<1/4 $相关的与扰动节点相关的非均匀离散傅立叶变换具有独立于$ H $的界限。我们继续表明，对于所有连续功能，不均匀间隔的正交规则会收敛，而interpolants均匀地收敛于所有可区分的功能，当$ 0 \ leqleqα<1/4 $时，其导数具有有限变化。但是，扰动节点处的正交规则对于任何$α> 0 $都可以具有负权重，我们提供了正交权重的绝对总和。因此，我们表明，没有数值的损失，可以使用带有小$α$的扰动均匀间隔的网格。虽然我们的证明技术主要是在$ 0 \leqα<1/4 $时起作用，但我们表明，少量的过采样将我们的结果扩展到$ 1/4 \ leqleqα<1/2 $时。
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 $1/4 \leq α< 1/2$.
