How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
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傅立叶矩阵的连续子矩阵的指数病态程度如何？
作者：
A. Barnett
我们证明了任何循环连续的$p的条件数 n的子矩阵 imes N$离散傅立叶变换（DFT）矩阵至少为$$ exp left（frac{pi}{2} left[min（p，q）- frac{pq}{N}
[8]
ight）~，$$直到代数前因子。也就是说，固定任何形状参数$（alpha，eta）：=（p/N，q/N）在（0，1）^2$中，增长是$e^{
ho N}$ as $N 美元，利率为$
h0 = frac{pi}{2}[min（alpha，eta）-
阿尔法eta]$。这样的范德蒙系统矩阵出现在许多应用中，例如傅立叶延拓、超分辨率和衍射成像。我们的证明使用Kaiser-Bessel变换对（我们给出了一个自包含的证明），并估计扭曲sinc函数的总和，以构建一个本地化的试验向量的DFT也本地化。我们通过一个周期化的高斯试验向量，用上述的一个基本证明来热身，但速度只有一半。使用核$e^{ixt}$的低秩近似，我们还证明了另一个下界$（4/epi alpha）^q$，直到代数前因子，对于小的$alpha，它比上面的更强， eta$。当结合起来时，边界在数值测量的经验渐近速率的两倍之内，一致超过$（0，1）^2$，并且在某些区域变得尖锐。然而，结果不是渐近的：它们适用于基本上所有的$N$，$p$和$q$，并且所有常数都是显式的。
We show that the condition number of any cyclically contiguous $p imes q$ submatrix of the $N imes N$ discrete Fourier transform (DFT) matrix is at least $$ exp left( frac{pi}{2} left[min(p,q)- frac{pq}{N}
ight]
ight)~, $$ up to algebraic prefactors. That is, fixing any shape parameters $(alpha,eta):=(p/N,q/N)in(0,1)^2$, the growth is $e^{
ho N}$ as $N oinfty$ with rate $
ho = frac{pi}{2}[min(alpha,eta)-
alphaeta]$. Such Vandermonde system matrices arise in many applications, such as Fourier continuation, super-resolution, and diffraction imaging. Our proof uses the Kaiser-Bessel transform pair (of which we give a self-contained proof), and estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kernel $e^{ixt}$, we also prove another lower bound $(4/epi alpha)^q$, up to algebraic prefactors, which is stronger than the above for small $alpha, eta$. When combined, the bounds are within a factor of two of the numerically-measured empirical asymptotic rate, uniformly over $(0,1)^2$, and they become sharp in certain regions. However, the results are not asymptotic: they apply to essentially all $N$, $p$, and $q$, and with all constants explicit.