Relevant sampling in finitely generated shift-invariant spaces

Abstract

We consider random sampling in finitely generated shift-invariant spaces V (Φ) ⊂ L 2 (R n ) generated by a vector Φ = (ϕ 1 , . . . , ϕ r ) ∈ L 2 (R n ) r . Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets V R,δ (Φ) of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths R. Under very mild assumptions on the generators, we show that for R sufficiently large, taking O(R n log(R n 2 /α ′ )) many random samples (taken independently uniformly distributed within C R ) yields a sampling set for V R,δ (Φ) with high probability. Here α ′ ≤ n is a suitable constant. We give explicit estimates of all involved constants in terms of the generators ϕ 1 , . . . , ϕ r .

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