Random sampling in shift invariant spaces

The set of sampling in a shift invariant space plays an important role in signal processing and has many applications. This paper addresses the problem when some randomly chosen samples X = { x j : j ∈ J } form a set of sampling in a shift invariant space. That is, when the inequality of the form c p ‖ f ‖ L p ( R d ) p ≤ ∑ x j ∈ X | f ( x j ) | p ≤ C p ‖ f ‖ L p ( R d ) p holds uniformly for all functions f in a shift invariant space, where c p and C p are positive constants ( 1 ≤ p ≤ ∞ ) . We prove that with overwhelming probability, the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large.