Uncertainty relations and sparse decompositions of analog signals

We consider uncertainty principles for analog signals that lie in a finitely-generated shift-invariant (SI) space. By adapting the notion of coherence defined for finite dictionaries to infinite SI representations, we develop an uncertainty principle similar in spirit to its finite counterpart. Building upon these results and similar work in the finite setting, we show how to find a sparse decomposition of an analog signal in an overcomplete dictionary by solving a convex optimization problem. The distinguishing feature of our approach is the fact that even though the problem is defined over an infinite domain with infinitely many variables and constraints, under certain conditions on the dictionary spectrum our algorithm can find the sparsest representation by solving a finite dimensional problem.