a b s t r a c tThe 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.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.
Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces [Formula: see text]. Under some decay conditions of the generator [Formula: see text], we approximate [Formula: see text] with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of [Formula: see text] when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by [Formula: see text]. In the end, we give a reconstruction algorithm for the random sampling of functions in [Formula: see text].
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