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Gelfand Widths of ℓ1-Balls

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

Abstract

This chapter makes a detour to the geometry of Banach spaces. First, it highlights a strong connection between compressive sensing and Gelfand widths, with implications on the minimal number of measurements needed for stable sparse recovery with an arbitrary measurement matrix. Then two-sided estimates for the Gelfand widths of 1-balls are established, as well as two-sided estimates for Kolmogorov widths. Finally, compressive sensing techniques are applied to give a proof of Kashin’s decomposition theorem.

Keywords

adaptive measurement scheme nonadaptive measurement scheme stability 1-minimization Gelfand width Kolmogorov width Kashin’s decomposition theorem 

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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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