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Ning Bi, M. Zuhair, Nashed And, Qiyu Sun (2009)
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We consider non-uniform random sampling in a signal space with finite rate of innovation V2(Λ,Φ)⊂L2(Rd)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$V^{2}(\varLambda,\varPhi) \subset{\mathrm {L}}^{2}(\mathbb {R}^{d})$\end{document} generated by a series of functions Φ=(ϕλ)λ∈Λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\varPhi=(\phi_{\lambda})_{\lambda \in\varLambda}$\end{document}. A subset VR,δ2(Λ,Φ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$V_{R,\delta}^{2}(\varLambda,\varPhi)$\end{document} of V2(Λ,Φ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$V^{2}(\varLambda,\varPhi)$\end{document} is consisting of functions concentrates at least 1−δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$1-\delta$\end{document} of the whole energy in a cube with side lengths R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$R$\end{document}. Under mild assumptions on the generators and the probability distribution, we show that for R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$R$\end{document} sufficiently large, taking O(Rdlog(Rd))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$O(R^{d} \log(R^{d}))$\end{document} many samples with such the non-uniform distribution yields a sampling set for VR,δ2(Λ,Φ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$V_{R,\delta}^{2}(\varLambda,\varPhi)$\end{document} with high probability. We impose compact support on the generators as an additional constraint for obtaining a reconstruction algorithm from non-uniform random sampling with high probability.
Acta Applicandae Mathematicae – Springer Journals
Published: Oct 6, 2020
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