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 $\mathcal{F}^{-1}h$ .