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Fusion Frames and Unbiased Basic Sequences

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Excursions in Harmonic Analysis, Volume 1

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

The construction of Parseval frames with special, rigid geometric properties has left many open problems even after decades of efforts. The construction of similar types of fusion frames is even less developed. We construct a large family of equi-isoclinic Parseval fusion frames by taking the Naimark complement of the union of orthonormal bases. If these bases are chosen to be mutually unbiased, then the resulting fusion frame subspaces are spanned by mutually unbiased basic sequences. By giving an explicit representation for Naimark complements, we are able to construct concrete fusion frames in their respective Hilbert spaces.

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Acknowledgements

Bernhard G. Bodmann was Supported by NSF DMS 0807399. Peter G. Casazza, Jesse D. Peterson, Ihar Smalyanau and Janet C. Tremain were Supported by NSF DMS 1008183, NSF ATD 1042701, and AFOSR FA9550-11-1-0245.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Bernhard G. Bodmann

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211-4100, USA

    Peter G. Casazza, Jesse D. Peterson, Ihar Smalyanau & Janet C. Tremain

Authors
  1. Bernhard G. Bodmann
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  2. Peter G. Casazza
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  3. Jesse D. Peterson
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  4. Ihar Smalyanau
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  5. Janet C. Tremain
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Corresponding author

Correspondence to Bernhard G. Bodmann .

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Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

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Bodmann, B.G., Casazza, P.G., Peterson, J.D., Smalyanau, I., Tremain, J.C. (2013). Fusion Frames and Unbiased Basic Sequences. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8376-4_2

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