Random Sampling in Bounded Orthonormal Systems

  • Simon Foucart
  • Holger Rauhut
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


This chapter considers the recovery of signals with a sparse expansion in a bounded orthonormal system. After an inventory of such bounded orthonormal systems including the Fourier systems, theoretical limitations specific to this situation are obtained for the minimal number of samples. Then, using this number of random samples, nonuniform sparse recovery is proved to be possible via ℓ1-minimization. Next, using a slightly higher number of random samples, uniform sparse recovery is also proved to be possible via a variety of algorithms. This is derived via establishing the restricted isometry property for the associated random sampling matrix—the random partial Fourier matrix is a particular case. Finally, a connection to the Λ 1-problem is pointed out.


sampling matrix random matrices bounded orthonormal systems discrete bounded orthonormal system partial Fourier matrix partial Hadamard matrix uncertainty principles nonuniform recovery 1-minimization golfing scheme restricted isometry property Λ1-problem 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Simon Foucart
    • 1
  • Holger Rauhut
    • 2
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Lehrstuhl C für Mathematik (Analysis)RWTH Aachen UniversityAachenGermany

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