Stable super-resolution limit and smallest singular value of restricted Fourier matrices

For many imaging tasks, it is necessary to locate a small number of point sources with high precision from their noisy low frequency Fourier coefficients. When the point sources are located on a fine grid, this problem is equivalent to recovering a N -dimensional S-sparse complex vector from noisy observations of its first M discrete Fourier coefficients, where N M and the noise level is bounded by δ. We show that whenM andN are sufficiently large, the min-max error is exactly δ C(S)(N/M)2S−1/ √ M , where C(S) is some constant depending only on S. This result is an immediate consequence of sharp lower and upper bounds on the smallest singular value of certain restricted Fourier matrices. We prove the lower bound using a duality characterization together with an interpolation argument involving the construction of a new family of trigonometric polynomials with small norms, which we call the sparse Lagrange polynomials. The upper bound can be interpreted as an extremal case of the discrete uncertainty principle.