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How Exponentially Ill-Conditioned Are Contiguous Submatrices of the Fourier Matrix?
SIAM Review ( IF 10.8 ) Pub Date : 2022-02-03 , DOI: 10.1137/20m1336837
Alex H. Barnett

SIAM Review, Volume 64, Issue 1, Page 105-131, February 2022.
Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT) matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such $p\times q$ submatrix of the $N\times N$ DFT matrix is at least $ \exp \left( \frac{\pi}{2} \left[\min(p,q)- \frac{pq}{N}\right]\right)$, up to algebraic prefactors. That is, fixing the shape parameters $(\al,\bt):=(p/N,q/N)\in(0,1)^2$, the growth is $e^{\rho N}$ as $N\to\infty$, the exponential rate being $\rho = \frac{\pi}{2}[\min(\alpha,\beta)- \alpha\beta]$. Our proof uses the Kaiser--Bessel transform pair (of which we give a self-contained proof), plus 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 $(4/e\pi \al)^q$, up to algebraic prefactors, which is stronger than the above for small $\al$ and $\bt$. When combined, the bounds are within a factor of two of the empirical asymptotic rate, uniformly over $(0,1)^2$, and 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.


中文翻译:

傅里叶矩阵的连续子矩阵的指数病态程度如何?

SIAM 评论,第 64 卷,第 1 期,第 105-131 页,2022 年 2 月。
涉及离散傅里叶变换 (DFT) 矩阵的连续子矩阵的线性系统出现在许多应用中,例如傅里叶扩展、超分辨率和相干衍射成像。我们证明了 $N\times N$ DFT 矩阵的任何这样的 $p\times q$ 子矩阵的条件数至少为 $ \exp \left( \frac{\pi}{2} \left[\min( p,q)- \frac{pq}{N}\right]\right)$,直到代数前置因子。即固定形状参数$(\al,\bt):=(p/N,q/N)\in(0,1)^2$,增长为$e^{\rho N}$为$N\to\infty$,指数速率为 $\rho = \frac{\pi}{2}[\min(\alpha,\beta)- \alpha\beta]$。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),加上对失真 sinc 函数的总和的估计,来构建一个局部化的试验向量,其 DFT 也是局部化的。我们通过周期化高斯试验向量对上述内容进行了初步证明,但速度减半。使用核$e^{ixt}$的低秩近似,我们还证明了另一个下界$(4/e\pi \al)^q$,直到代数预因子,这比上面的小$更强\al$ 和 $\bt$。结合起来,边界在经验渐近率的两倍以内,一致地超过 $(0,1)^2$,并且在某些区域变得尖锐。然而,结果不是渐近的:它们基本上适用于所有 $N$、$p$ 和 $q$,并且所有常量都是显式的。结合起来,边界在经验渐近率的两倍以内,一致地超过 $(0,1)^2$,并且在某些区域变得尖锐。然而,结果不是渐近的:它们基本上适用于所有 $N$、$p$ 和 $q$,并且所有常量都是显式的。结合起来,边界在经验渐近率的两倍以内,一致地超过 $(0,1)^2$,并且在某些区域变得尖锐。然而,结果不是渐近的:它们基本上适用于所有 $N$、$p$ 和 $q$,并且所有常量都是显式的。
更新日期:2022-02-03
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