Frames Containing a Riesz Basis and Approximation of the Inverse Frame Operator

  • Ole Christensen
  • Alexander Lindner
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 137)

Abstract

A frame in a Hilbert space H allows every element in H to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of those coefficients and many other situations where frames occur, require knowledge of the inverse frame operator. But usually it is hard to invert the frame operator if the underlying Hilbert space is infinite dimensional. We introduce a method for approximation of the inverse frame operator using finite subsets of the frame. In particular this allows to approximate the frame coefficients (even in 2—sense) using finite-dimensional linear algebra. We show that the general method simplifies when the frame contains a Riesz basis.

Keywords

Riesz Basis Wavelet Frame Gabor Frame Finite Family Lower Frame 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Casazza, O. Christensen: Frames containing a Riesz basis and preservation of this property under perturbation. SIAM J. Math. Anal. 29 (1998), 266–278.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    O. Christensen: Frames containing a Riesz basis and approxima- tion of the frame coefficients using finite dimensional methods, J. Math. Anal. Appl. 199 (1996), 256–270.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    O. Christensen: Finite-dimensional approximation of the inverse frame operator and applications to Weyl-Heisenberg frames and wavelet frames, J. Fourier Anal. Appl. 6 (2000), 79–91.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    O. Christensen, A. Lindner: Lower frame bounds for finite wavelet and Gabor systems, Approx. Theory and its Appl., to appear.Google Scholar
  5. [5]
    O. Christensen, A. Lindner: Frames of exponentials: lower frame bounds for finite subfamilies, and approximation of the inverse frame operator, Linear Alg. Appl. 323 (2001), 117–130.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    R. J. Duffin, A. C. Schaeffer: A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    K. Seip: On the connection between exponential bases and certain related sequences in L 2(-π,π), J. Funct. Anal. 130 (1995), 131–160.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    R. Young: An Introduction to Nonharmonic Fourier Series, Aca-demic Press, New York, 1980.MATHGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Ole Christensen
    • 1
  • Alexander Lindner
    • 2
  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark
  2. 2.Fachbereich MathematikTechnische Universität MünchenMünchenGermany

Personalised recommendations