Sparse Legendre expansions via l1-minimization

  • Authors:
  • Holger Rauhut;Rachel Ward

  • Affiliations:
  • Hausdorff Center for Mathematics & Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany;Mathematics Department, University of Texas at Austin, 2515 Speedway, RLM 10.144, Austin, TX, 78712, USA

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2012

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Abstract

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered from m@?slog^4(N) random samples that are chosen independently according to the Chebyshev probability measure d@n(x)=@p^-^1(1-x^2)^-^1^/^2dx. As an efficient recovery method, @?"1-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces.