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Constructing Finite Frames with a Given Spectrum

  • Chapter
Finite Frames

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

Abstract

Broadly speaking, frame theory is the study of how to produce well-conditioned frame operators, often subject to nonlinear application-motivated restrictions on the frame vectors themselves. In this chapter, we focus on one particularly well-studied type of restriction: having frame vectors of prescribed lengths. We discuss two methods for iteratively constructing such frames. The first method, called Spectral Tetris, produces special examples of such frames, and only works in certain cases. The second method combines the idea behind Spectral Tetris with the classical theory of majorization; this method can build any such frame in terms of a sequence of interlacing spectra, called eigensteps.

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Acknowledgements

This work was supported by NSF DMS 1042701, NSF CCF 1017278, AFOSR F1ATA01103J001, AFOSR F1ATA00183G003, and the A.B. Krongard Fellowship. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, the U.S. Government, or Thomas Jefferson.

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Authors and Affiliations

  1. Department of Mathematics, Air Force Institute of Technology, Wright-Patterson AFB, OH, 45433, USA

    Matthew Fickus & Miriam J. Poteet

  2. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Dustin G. Mixon

Authors
  1. Matthew Fickus
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  2. Dustin G. Mixon
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  3. Miriam J. Poteet
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Correspondence to Matthew Fickus .

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Editors and Affiliations

  1. , Department of Mathematics, University of Missouri, 202 Mathematical Sciences Building, Columbia, 65211, Missouri, USA

    Peter G. Casazza

  2. , Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Gitta Kutyniok

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Fickus, M., Mixon, D.G., Poteet, M.J. (2013). Constructing Finite Frames with a Given Spectrum. In: Casazza, P., Kutyniok, G. (eds) Finite Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8373-3_2

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