Convolution random sampling in multiply generated shift-invariant spaces of Lp(Rd)

We mainly consider the stability and reconstruction of convolution random sampling in multiply generated shift-invariant subspaces V p (Φ) = ∑ k ∈ Z d c (k) T Φ (· - k) : (c (k)) k ∈ Z d ∈ (ℓ p (Z d)) r <graphic href="43034_2020_98_Article_Equ51.gif"></graphic> of L p (R d) , 1 < p < ∞ , where Φ = (ϕ 1 , ϕ 2 , … , ϕ r) T with ϕ i ∈ L p (R d) and c = (c 1 , c 2 , … , c r) T with c i ∈ ℓ p (Z d) , i = 1 , 2 , … , r . The sampling set { x j } j ∈ N is randomly chosen with a general probability distribution over a bounded cube C K and the samples are the form of convolution { f ∗ ψ (x j) } j ∈ N of the signal f. Under some proper conditions for the generator Φ , convolution function ψ and probability density function ρ , we first approximate V p (Φ) by a finite dimensional subspace V N p (Φ) = ∑ i = 1 r ∑ | k | ≤ N c i (k) ϕ i (· - k) : c i ∈ ℓ p ([ - N , N ] d). <graphic href="43034_2020_98_Article_Equ52.gif"></graphic> Then we show that the sampling stability holds with high probability for all functions in certain compact subsets V K p (Φ) = f ∈ V p (Φ) : ∫ C K | f ∗ ψ (x) | p d x ≥ (1 - δ) ∫ R d | f ∗ ψ (x) | p d x <graphic href="43034_2020_98_Article_Equ53.gif"></graphic> of V p (Φ) when the sampling size is large enough. Finally, we prove that the stability is related to the properties of the random matrix generated by { ϕ i ∗ ψ } 1 ≤ i ≤ r and give a reconstruction algorithm for the convolution random sampling of functions in V N p (Φ) . [ABSTRACT FROM AUTHOR]