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 Grochenig, 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 l o g ( R ) ) many random samples (taken independently uniformly distributed within C R ) yields a sampling set for V R , δ ( Φ ) with high probability. We give explicit estimates of all involved constants in terms of the generators φ 1 , … , φ r .