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Finite Frames pp 171–191Cite as

Birkhäuser

Group Frames

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

Abstract

The prototypical example of a tight frame, the Mercedes-Benz frame can be obtained as the orbit of a single vector under the action of the group generated by rotation by \(\frac{2\pi}{3}\), or the dihedral group of symmetries of the triangle. Many frames used in applications are constructed in this way, often as the orbit of a single vector (akin to a mother wavelet). Most notable are the harmonic frames (finite abelian groups) used in signal analysis, and the equiangular Heisenberg frames, or SIC–POVMs (discrete Heisenberg group) used in quantum information theory. Other examples include tight frames of multivariate orthogonal polynomials sharing symmetries of the weight function, and the highly symmetric tight frames which can be viewed as the vertices of highly regular polytopes. We will describe the basic theory of such group frames, and some of the constructions that have been found so far.

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References

  1. Appleby, D.M., Dang, H.B., Fuchs, C.A.: Symmetric informationally-complete quantum states as analogues to orthonormal bases and minimum-uncertainty states (2007). arXiv:0707.2071v2 [quant-ph]

  2. Bos, L., Waldron, S.: Some remarks on Heisenberg frames and sets of equiangular lines. N.Z. J. Math. 36, 113–137 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Brauer, R., Coxeter, H.S.M.: A generalization of theorems of Schöhardt and Mehmke on polytopes. Trans. R. Soc. Canada Sect. III 34, 29–34 (1940)

    MathSciNet  Google Scholar 

  4. Broome, H., Waldron, S.: On the construction of highly symmetric tight frames and complex polytopes. Preprint (2010)

    Google Scholar 

  5. Chien, T., Waldron, S.: A classification of the harmonic frames up to unitary equivalence. Appl. Comput. Harmon. Anal. 30, 307–318 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coxeter, H.S.M.: Regular Complex Polytopes. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  7. Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Goyal, V.K., Vetterli, M., Thao, N.T.: Quantized overcomplete expansions in ℝn: analysis, synthesis, and algorithms. IEEE Trans. Inf. Theory 44, 16–31 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hay, N., Waldron, S.: On computing all harmonic frames of n vectors in ℂd. Appl. Comput. Harmon. Anal. 21, 168–181 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hochwald, B., Marzetta, T., Richardson, T., Sweldens, W., Urbanke, R.: Systematic design of unitary space-time constellations. IEEE Trans. Inf. Theory 46, 1962–1973 (2000)

    Article  MATH  Google Scholar 

  12. James, G., Liebeck, M.: Representations and Characters of Groups. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  13. Kalra, D.: Complex equiangular cyclic frames and erasures. Linear Algebra Appl. 419, 373–399 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khatirinejad, M.: On Weyl-Heisenberg orbits of equiangular lines. J. Algebr. Comb. 28, 333–349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Renes, J.M., Blume–Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Scott, A.J., Grassl, M.: SIC-POVMs: A new computer study (2009). arXiv:0910.5784v2 [quant-ph]

  17. Vale, R., Waldron, S.: Tight frames and their symmetries. Constr. Approx. 21, 83–112 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Vale, R., Waldron, S.: Tight frames generated by finite nonabelian groups. Numer. Algorithms 48, 11–27 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vale, R., Waldron, S.: The symmetry group of a finite frame. Linear Algebra Appl. 433, 248–262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Waldron, S.: An Introduction to Finite Tight Frames. Springer, New York (2011)

    Google Scholar 

  21. Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51, 1900–1907 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zauner, G.: Quantendesigns—Grundzüge einer nichtkommutativen Designtheorie. Doctorial thesis, University of Vienna, Vienna, Austria (1999)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Private Bag, 92019, Auckland, New Zealand

    Shayne Waldron

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  1. Shayne Waldron
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Correspondence to Shayne Waldron .

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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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Waldron, S. (2013). Group 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_5

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