  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 is a prime power, we characterize all such h with a prescribed
zero set in terms of digit expansions of nonzero indices in the inverse DFT of
h.
