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Equiangular Frames and Their Duals

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Excursions in Harmonic Analysis, Volume 6

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

Abstract

Systems of m equiangular lines spanning \(\mathbb {R}^d\) or \(\mathbb {C}^d\) that satisfy the so-called Welch bound have recently gained a lot of attention due to various applications in signal processing. Such sets are called equiangular tight frames (ETFs). One of the geometrically appealing aspects of an ETF is that any vector can be represented in terms of an ETF by using a dual frame that is also an equiangular set. However, for a given m and d, with m > d + 1, ETFs are rare. Here we study some properties of equiangular lines spanning \(\mathbb {R}^d\) when the Welch bound is not met. Such equiangular sets are more common than ETFs. In this case, the properties of the canonical dual, in particular, the angle set of the canonical dual are studied. We determine conditions on equiangular lines spanning \(\mathbb {R}^d\) whose canonical dual has few distinct angles.

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References

  1. Bodmann, B.G., Elwood, H.J.: Complex equiangular Parseval frames and Seidel matrices containing pth roots of unity. Proc. Amer. Math. Soc. 138(12), 4387–4404 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bodmann, B.G., Paulsen, V.I., Tomforde, M.: Equiangular tight frames from complex Seidel matrices containing cube roots of unity. Linear Algebra Appl., 430 (1), 396–417 (2009)

    Article  MathSciNet  Google Scholar 

  3. Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl., 195 (2), 401–414 (1995)

    Article  MathSciNet  Google Scholar 

  4. Christensen, O.: An Introduction to Frames and Riesz Bases. Second expanded edition, Birkhäuser, Boston, Basel, Berlin (2016)

    Google Scholar 

  5. Christensen, O., Datta, S., Kim, R.Y.: Equiangular frames and generalizations of the Welch bound to dual pairs of frames. Linear Multilinear Algebra. 68(12), 2495–2505 (2020)

    Article  MathSciNet  Google Scholar 

  6. Christensen, O., Datta, S., Kim, R.Y.: Duals and approximate duals of equiangular frames. In preparation.

    Google Scholar 

  7. Christensen, O., Powell, A., Xiao, X. C.: A note on finite dual frame pairs. Proc. Amer. Math. Soc. 140 (11), 3921–3930 (2012)

    Article  MathSciNet  Google Scholar 

  8. Datta, S., Oldroyd, J.: Construction of k-angle tight frames. Numer. Funct. Anal. Optim. 37 (8), 975–989 (2016)

    Article  MathSciNet  Google Scholar 

  9. Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30 (3) Issue in honour of C.J. Bouwkamp, 91–105 (1975)

    Google Scholar 

  10. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72 (2), 341–366 (1952)

    Article  MathSciNet  Google Scholar 

  11. Fickus, M., Mixon, D.G., Tremain, J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436(5), 1014–1027 (2012)

    Article  MathSciNet  Google Scholar 

  12. Greaves, G.: Equiangular lines systems and switching classes containing regular graphs. Linear Algebra Appl. 536, 31–51 (2018)

    Article  MathSciNet  Google Scholar 

  13. Greaves, G., Koolen, J.H., Munemasa, A., Szöllősi, F.: Equiangular lines in Euclidean spaces. J. Combin. Theory Ser. A. 138, 208–235 (2016)

    Google Scholar 

  14. Hoggar, S.G.: t-designs in projective spaces. European J. Combin. 3 (3), 233–254 (1982)

    Google Scholar 

  15. Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31-51 (2004)

    Article  MathSciNet  Google Scholar 

  16. Klappenecker, A., Rötteler, M.: Mutually unbiased bases are complex projective 2-designs. In: Proceedings of the International Symposium on Information Theory, pp. 1740–1744, (2005)

    Google Scholar 

  17. Lemmens,P.W.H, Seidel, J.J.: Equiangular lines. J. Algebra. 24 (3), 494–512 (1973)

    Article  MathSciNet  Google Scholar 

  18. Massey, J.L., Mittelholzer, T.: Welch’s bound and sequence sets for code-division multiple-access systems. In: Capocelli, R., De Santis, A., Vaccaro, U. (eds) Sequences II: Methods in Communication, Security and Computer Science, pp. 63–78. Springer-Verlag, New York (1991)

    Google Scholar 

  19. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)

    Article  MathSciNet  Google Scholar 

  20. Roy, A., Scott, A.J.: Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements. J. Math. Phys. 48 (072110), (2007)

    Google Scholar 

  21. Sarwate, D.V.: Meeting the Welch bound with equality. In: Helleseth, T., Niederreiter, H.(eds) Sequences and Their Applications, DMTCS Series, chapter11. Springer-Verlag (1999)

    Google Scholar 

  22. Scott, A.J: Tight informationally complete quantum measurements. J. Phys. A 39, 13507–13530 (2006)

    Article  MathSciNet  Google Scholar 

  23. Strohmer, T., Heath Jr., R.W.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)

    Article  MathSciNet  Google Scholar 

  24. Sustik, M.A., Tropp, J.A., Dhillon, I.S., Heath Jr., R.W.: On the existence of equiangular tight frames. Linear Algebra Appl. 426(2-3), 619–635 (2007)

    Article  MathSciNet  Google Scholar 

  25. Szöllősi, F., Östergård, R. J.: Enumeration of Seidel matrices. European J. Combin. 69, 169–184 (2018)

    Google Scholar 

  26. Tóth, L.F.: Distribution of points in the elliptic plane. Acta Math. Hungar. 16, 437–440 (1965)

    Article  MathSciNet  Google Scholar 

  27. Tropp, J.A.: Complex equiangular tight frames. In: Proceedings of SPIE 2005 (Wavelets XI), pp. 590412.01–590412.11. (2005)

    Google Scholar 

  28. Vale, R., Waldron, S.: Tight frames and their symmetries. Constr. Approx. 21 (1), 83–112 (2004)

    MathSciNet  MATH  Google Scholar 

  29. van Dam, E.R.: Regular graphs with four eigenvalues. Linear Algebra Appl. 226/228, 139–162 (1995)

    Google Scholar 

  30. Waldron, S.: On the construction of equiangular frames from graphs. Linear Algebra Appl. 431 (11), 2228–2242 (2009)

    Article  MathSciNet  Google Scholar 

  31. Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory 20 (3), 397–399 (May 1974)

    Article  Google Scholar 

  32. Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inform. Theory 51 (5), 1900–1907 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful to Prof. Ole Christensen for many useful discussions, and to the Department of Applied Mathematics and Computer Science, Technical University of Denmark for hospitality during a visit when some of this work was done. The author would like to thank Dr. Rae Young Kim for providing many helpful comments on this work. Finally, the author wishes to thank the anonymous reviewer for his/her thorough efforts in reading and commenting on the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Idaho, Moscow, ID, USA

    Somantika Datta

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  1. Somantika Datta
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Correspondence to Somantika Datta .

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Editors and Affiliations

  1. Department of Computational Mathematics, Science & Engineering, Michigan State University, East Lansing, MI, USA

    Matthew Hirn

  2. Department of Mathematics, San Francisco State University, San Francisco, CA, USA

    Shidong Li

  3. Department of Mathematics, Tufts University, Medford, MA, USA

    Kasso A. Okoudjou

  4. Department of Mathematics, Computer Science and Economics, University of Basilicata, Potenza, Potenza, Italy

    Sandra Saliani

  5. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Özgür Yilmaz

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Datta, S. (2021). Equiangular Frames and Their Duals. In: Hirn, M., Li, S., Okoudjou, K.A., Saliani, S., Yilmaz, Ö. (eds) Excursions in Harmonic Analysis, Volume 6. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-69637-5_9

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