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Algebraic Geometry and Finite Frames

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Finite Frames

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

Abstract

Interesting families of finite frames often admit characterizations in terms of algebraic constraints, and thus it is not entirely surprising that powerful results in finite frame theory can be obtained by utilizing tools from algebraic geometry. In this chapter, our goal is to demonstrate the power of these techniques. First, we demonstrate that algebro-geometric ideas can be used to explicitly construct local coordinate systems that reflect intuitive degrees of freedom within spaces of finite unit norm tight frames (and more general spaces), and that optimal frames can be characterized by useful algebraic conditions. In particular, we construct locally well-defined real-analytic coordinate systems on spaces of finite unit norm tight frames, and we demonstrate that many types of optimal Parseval frames are dense and that further optimality can be discovered through embeddings that naturally arise in algebraic geometry.

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Author information

Authors and Affiliations

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

    Jameson Cahill

  2. Department of Mathematics, Duke University, Durham, NC, USA

    Nate Strawn

Authors
  1. Jameson Cahill
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  2. Nate Strawn
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Correspondence to Nate Strawn .

Editor information

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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Cahill, J., Strawn, N. (2013). Algebraic Geometry and Finite Frames. 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_4

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