The Smallest Singular Value of Nonuniform Fourier Matrices

We study the smallest singular value of nonuniform Fourier matrices in two settings: clustered nodes and perturbations of an equispaced grid. By reducing the problem to spectral norm estimates for periodic nonuniform interpolation matrices, we obtain nearly optimal bounds in both cases. For clustered nodes, we derive a local separation condition in which each required gap depends only on the sizes of the two neighboring clusters. For perturbations with the bound $1\mathrm{/}4\le L<1\mathrm{/}2$, our result confirms the conjecture of Austin and Trefethen on the $2$-norm Lebesgue constant up to a logarithmic factor.