How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
share
We show that the condition number of any cyclically contiguous p× q submatrix of the N× N discrete Fourier transform (DFT) matrix is at least ( π/2[min(p,q)- pq/N] ) , up to algebraic prefactors. That is, fixing any shape parameters (α,β):=(p/N,q/N)∈(0,1)^2, the growth is e^ρ N as N→∞ with rate ρ = π/2[min(α,β)- αβ]. Such Vandermonde system matrices arise in Fourier continuation, super-resolution, and diffraction imaging. Our proof uses the Kaiser-Bessel transform pair, 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, our bounds appear to be within a factor of 2 of the numerically-measured empirical asymptotic rate, uniformly over (0,1)^2, and they become sharp in certain regions. However, our results are not asymptotic: they apply to essentially all N, p, and q, and with all constants explicit.
READ FULL TEXT