LP relaxations and Fuglede's conjecture

Computing the Discrete Fourier Transform of signals with spectral frequency support

“…Thus, the zero set contains all the indices in Z N whose gcd with N is in D J . The proof is elementary: we direct the interested reader to the references ( [24, Theorem 2.1], [9], [12], [25]) for the proof and details. We refer to the set D J often as zero-set divisors of h J .…”

“…Indeed, the periodic and consecutive element sets mentioned earlier are examples of spectral sets. These sets are relevant in the context of Fuglede's conjecture [7] in Fourier analysis, and this conjecture is as yet open for the discrete case for arbitrary N [8], [9].…”

Convolution Idempotents with a given Zero-set

“…See Figure 1 for an example signal with F = {0, 2} (i.e. 2 fragments), with the 0 th fragment in [0, 1] and 2 nd fragment in [2,3]. Regular sampling of the signals in S at the Nyquist-Shannon sampling rate [7] might not make use of gaps in the fragmented spectrum, with the consequence that the sampling rate could be higher than required for reconstructing the signal.…”

“…and so h J satisfies the hypothesis of Proposition 2. Indeed, the spectrum of the sampled signal has the form The original fragments in the intervals [0, 1] and [2,3] are unchanged, but the gaps between fragments are filled with aliases. We have a few more comments.…”

Some results on convolution idempotents

“…for some set of divisors D(h) ⊆ D N . We call D(h) the zero-set divisors of h. The lemma appears in many different forms and contexts, see [5], [6], [7] or [8, Theorem 2.1] for example: the key ingredient in the proof is the structure of cyclotomic polynomials.…”