We mainly consider the stability and reconstruction of convolution random sampling in multiply generated shift-invariant subspaces Vp(Φ)=∑k∈Zdc(k)TΦ(·-k):(c(k))k∈Zd∈(ℓp(Zd))r\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} V^{p}(\varPhi )=\left\{ \sum \limits _{k\in \mathbb {Z}^{d}}c(k)^{T}\varPhi (\cdot -k):(c(k))_{k\in \mathbb {Z}^{d}}\in (\ell ^{p}(\mathbb {Z}^{d}))^r \right\} \end{aligned}$$\end{document}of Lp(Rd)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$L^p(\mathbb {R}^{d})$$\end{document}, 1<p<∞\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1<p<\infty$$\end{document}, where Φ=(ϕ1,ϕ2,…,ϕr)T\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\varPhi =(\phi _{1},\phi _{2},\ldots ,\phi _{r})^{T}$$\end{document} with ϕi∈Lp(Rd)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi _{i}\in L^{p}(\mathbb {R}^{d})$$\end{document} and c=(c1,c2,…,cr)T\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c=(c_{1},c_{2},\ldots , c_{r})^{T}$$\end{document} with ci∈ℓp(Zd)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c_{i}\in \ell ^{p}(\mathbb {Z}^{d})$$\end{document}, i=1,2,…,r\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$i=1,2,\ldots , r$$\end{document}. The sampling set {xj}j∈N\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\{x_j\}_{j\in \mathbb {N}}$$\end{document} is randomly chosen with a general probability distribution over a bounded cube CK\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$C_{K}$$\end{document} and the samples are the form of convolution {f∗ψ(xj)}j∈N\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\{f*\psi (x_j)\}_{j\in \mathbb {N}}$$\end{document} of the signal f. Under some proper conditions for the generator Φ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\varPhi$$\end{document}, convolution function ψ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\psi$$\end{document} and probability density function ρ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho$$\end{document}, we first approximate Vp(Φ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$V^{p}(\varPhi )$$\end{document} by a finite dimensional subspace VNp(Φ)=∑i=1r∑|k|≤Nci(k)ϕi(·-k):ci∈ℓp([-N,N]d).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} V^{p}_{N}(\varPhi )=\left\{ \sum \limits _{i=1}^{r}\sum \limits _{|k|\le N}c_{i}(k)\phi _{i}(\cdot -k): c_{i}\in \ell ^{p}([-N,N]^{d})\right\} . \end{aligned}$$\end{document}Then we show that the sampling stability holds with high probability for all functions in certain compact subsets VKp(Φ)=f∈Vp(Φ):∫CK|f∗ψ(x)|pdx≥(1-δ)∫Rd|f∗ψ(x)|pdx\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} V^{p}_{K}(\varPhi )=\left\{ f\in V^{p}(\varPhi ):\int _{C_{K}}|f*\psi (x)|^{p}dx\ge (1-\delta )\int _{\mathbb {R}^{d}}|f*\psi (x)|^{p}dx\right\} \end{aligned}$$\end{document}of Vp(Φ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$V^{p}(\varPhi )$$\end{document} when the sampling size is large enough. Finally, we prove that the stability is related to the properties of the random matrix generated by {ϕi∗ψ}1≤i≤r\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\{\phi _i*\psi \}_{1\le i\le r}$$\end{document} and give a reconstruction algorithm for the convolution random sampling of functions in VNp(Φ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$V^{p}_N(\varPhi )$$\end{document}.
