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A generalization of Gram–Schmidt orthogonalization generating all Parseval frames

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Abstract

Given an arbitrary finite sequence of vectors in a finite-dimensional Hilbert space, we describe an algorithm, which computes a Parseval frame for the subspace generated by the input vectors while preserving redundancy exactly. We further investigate several of its properties. Finally, we apply the algorithm to several numerical examples.

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References

  1. J.J. Benedetto and M. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003) 357–385.

    Article  MATH  MathSciNet  Google Scholar 

  2. P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000) 129–201.

    MATH  MathSciNet  Google Scholar 

  3. P.G. Casazza, Custom building finite frames, in: Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (Amer. Math. Soc., Providence, RI, 2004) pp. 61–86.

    Google Scholar 

  4. P.G. Casazza and J. Kovaĉević, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003) 387–430.

    Article  MATH  MathSciNet  Google Scholar 

  5. O. Christensen, An Introduction to Frames and Riesz Bases (Birkhäuser, Boston, 2003).

    MATH  Google Scholar 

  6. V.K. Goyal, J. Kovačević and J.A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal. 10 (2001) 203–233.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Peter G. Casazza

  2. Mathematical Institute, Justus-Liebig-University Giessen, 35392, Giessen, Germany

    Gitta Kutyniok

Authors
  1. Peter G. Casazza
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  2. Gitta Kutyniok
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Corresponding author

Correspondence to Peter G. Casazza.

Additional information

Communicated by J.J. Benedetto

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Casazza, P.G., Kutyniok, G. A generalization of Gram–Schmidt orthogonalization generating all Parseval frames. Adv Comput Math 27, 65–78 (2007). https://doi.org/10.1007/s10444-005-7478-1

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  • DOI: https://doi.org/10.1007/s10444-005-7478-1

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