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Finite Frames pp 141-170 | Cite as

Algebraic Geometry and Finite Frames

Part of the Applied and Numerical Harmonic Analysis book series (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.

Keywords

Algebraic geometry Elimination theory Plücker embedding Finite frames 

References

  1. 1.
    Balan, R.V.: Equivalence relations and distances between Hilbert frames. Proc. Am. Math. Soc. 127, 2353–2366 (1999) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Balan, R.V., Casazza, P.G., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20, 345–356 (2006) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Benedetto, J.J., Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18, 357–385 (2003) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  5. 5.
    Cahill, J.: Flags, frames, and Bergman spaces. Master’s Thesis, San Francisco State University (2009) Google Scholar
  6. 6.
    Cahill, J., Casazza, P.G.: The Paulsen problem in operator theory (2011). arXiv:1102.2344
  7. 7.
    Casazza, P.G., Leon, M.T.: Existence and construction of finite frames with a given frame operator. Int. J. Pure Appl. Math. 63, 149–158 (2010) MathSciNetMATHGoogle Scholar
  8. 8.
    Casazza, P.G., Tremain, J.C.: The Kadison–Singer problem in mathematics and engineering. Proc. Natl. Acad. Sci. 103, 2032–2039 (2006) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dykema, K., Strawn, N.: Manifold structure of spaces of spherical tight frames. Int. J. Pure Appl. Math. 28, 217–256 (2006) MathSciNetMATHGoogle Scholar
  10. 10.
    Fraenkel, A.S., Yesha, Y.: Complexity of problems in games, graphs, and algebraic equations. Discrete Appl. Math. 1, 15–30 (1979) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fulton, W.: Young Tableaux—With Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  12. 12.
    Guillemin, V., Pollack, A.: Differential Topology—History, Theory, and Applications. Prentice-Hall, Englewood Cliffs (1974) Google Scholar
  13. 13.
    Hartshorne, R.: Algebraic Geometry. Springer, New York (1997) Google Scholar
  14. 14.
    Jiang, S.: Angles between Euclidean subspaces. Geom. Dedic. 63, 113–121 (1996) MATHCrossRefGoogle Scholar
  15. 15.
    Jordan, C.: Essai sur la géométrie á n dimensions. Bull. Soc. Math. Fr. 3, 103–174 (1875) MATHGoogle Scholar
  16. 16.
    Krantz, S.G., Parks, H.R.: The Implicit Function Theorem—History, Theory, and Applications. Birkhäuser, Boston (2002) MATHCrossRefGoogle Scholar
  17. 17.
    Miao, J.M., Ben-Israel, A.: On principal angles between subspaces in ℝn. Linear Algebra Appl. 171, 81–98 (1992) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Miao, J.M., Ben-Israel, A.: Product cosines of angles between subspaces. Linear Algebra Appl, 237–238:71–81 (1996) Google Scholar
  19. 19.
    Oxley, J.G.: Matroid Theory. Oxford University Press, New York (1992) MATHGoogle Scholar
  20. 20.
    Püschel, M., Kovačević, J.: Real tight frames with maximal robustness to erasures. In: Proc. IEEE Data Comput. Conf., pp. 63–72 (2005) Google Scholar
  21. 21.
    Strawn, N.: Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. J. Fourier Anal. Appl. 17, 821–853 (2011) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Weaver, N.: The Kadison–Singer problem in discrepancy theory. Discrete Math. 278, 227–239 (2004) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA

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