Convolution Idempotents with a given Zero-set

TL;DR

This paper characterizes discrete signals that are idempotent under convolution and vanish on a specified set, linking their structure to digit expansions of inverse DFT indices, with applications in sampling and bandlimited signal interpolation.

Contribution

It provides a complete characterization of convolution idempotents with prescribed zeros for prime power lengths using digit expansion techniques.

Findings

Characterization of idempotent signals with zero constraints for prime power lengths

Connection between zero sets and digit expansions of inverse DFT indices

Applications to sampling and orthogonal interpolation of bandlimited signals

Abstract

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.