论文标题
傅立叶矩阵的连续一键率呈指数状态？
How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
论文作者
论文摘要
我们表明，$ n \ times n $ n $离散傅里叶变换（DFT）矩阵的任何周期性连续$ p \ times q $ subsrix的条件数至少为$$ \ exp \ weft（\fracπ{2} \ left [\ left [\ min（p，p，q） - \ frac {pq}预成分。也就是说，固定任何形状参数$（α，β）：=（p/n，q/q/q/n）\ in（0,1）^2 $，生长为$ e^{ρn} $作为$ n \ to \ infty $，to \ infty $，rate $ρ= \fracπ{2} [2} [2} [\ min（α，α） - α，α-α，α，α，α，α，α， - αβ] $。这种Vandermonde系统矩阵在许多应用中都出现，例如傅立叶延续，超分辨率和衍射成像。我们的证明使用Kaiser-Bessel变换对（我们提供了独立的证明），并对SINC函数的总和进行了估算，以构建DFT的局部试验向量，其DFT也是本地化的。我们用上述基本证据进行热身，但通过周期的高斯试验矢量进行了一半的速度。使用内核$ e^{ixt} $的低级别近似，我们还证明了另一个下限$（4/eπα）^q $，直至代数预先成分，该代数比上面的$α，β$要强。合并后，边界在数值测量的经验渐近率的两个因子内，均匀地超过$（0,1）^2 $，并且它们在某些地区变得敏锐。但是，结果不是渐近的：它们基本上适用于所有$ n $，$ p $和$ q $，并且明确适用于所有常数。
We show that the condition number of any cyclically contiguous $p\times q$ submatrix of the $N\times N$ discrete Fourier transform (DFT) matrix is at least $$ \exp \left( \fracπ{2} \left[\min(p,q)- \frac{pq}{N}\right] \right)~, $$ up to algebraic prefactors. That is, fixing any shape parameters $(α,β):=(p/N,q/N)\in(0,1)^2$, the growth is $e^{ρN}$ as $N\to\infty$ with rate $ρ= \fracπ{2}[\min(α,β)- αβ]$. 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 $(4/eπα)^q$, up to algebraic prefactors, which is stronger than the above for small $α, β$. 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.
