On sampling theorem with sparse decimated samples: exploring branching spectrum degeneracy

Abstract: The paper investigates possibility of recovery of sequences from their decimated subsequences. It is shown that this recoverability is associated with certain spectrum degeneracy of a new kind, and that a sequences of a general kind can be approximated by sequences featuring this degeneracy. This is applied to sparse sampling of continuous time band-limited functions. The paper shows that these functions allow an arbitrarily close approximation by functions that can be recovered from sparse equidistant samples with sampling distance larger than the distance defined by the critical Nyquist rate for the underlying function. This allows to bypass, in a certain sense, the restriction on the sampling rate defined by the Nyquist rate.