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• W3019374509 abstract "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 $ptimes q$ submatrix of the $Ntimes N$ 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 $Ntoinfty$, the exponential rate being $rho = frac{pi}{2}[min(alpha,beta)- alphabeta]$. 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 $(4/epi al)^q$, 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." @default.
• W3019374509 created "2020-05-01" @default.
• W3019374509 creator A5037306695 @default.
• W3019374509 date "2022-02-01" @default.
• W3019374509 modified "2023-09-26" @default.
• W3019374509 title "How Exponentially Ill-Conditioned Are Contiguous Submatrices of the Fourier Matrix?" @default.
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• W3019374509 doi "https://doi.org/10.1137/20m1336837" @default.
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