  We study the problem of super-resolution of a linear combination of Dirac
distributions and their derivatives on a one-dimensional circle from noisy
Fourier measurements. Following numerous recent works on the subject, we
consider the geometric setting of "partial clustering", when some Diracs can be
separated much below the Rayleigh limit. Under this assumption, we prove sharp
asymptotic bounds for the smallest singular value of a corresponding
rectangular confluent Vandermonde matrix with nodes on the unit circle. As a
consequence, we derive matching lower and upper min-max error bounds for the
above super-resolution problem, under the additional assumption of nodes
belonging to a fixed grid.
