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={xj:j∈J} form a set of sampling in a shift invariant space. That is, when the inequality of the form cp‖f‖Lp(Rd)p≤∑xj∈X|f(xj)|p≤Cp‖f‖Lp(Rd)p holds uniformly for all functions f in a shift invariant space, where cp and Cp 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.