Convolution Idempotents With a Given Zero-Set

We investigate the structure of N-length discrete signals h satisfying h * h = h that vanish on a given set of indices. We motivate this problem from examples in sampling, Fuglede's conjecture, and orthogonal interpolation of bandlimited signals. When N = p M is a prime power, we characterize all such h with a prescribed zero set in terms of base-p expansions of nonzero indices in F -1 h.