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The HRT Conjecture and the Zero Divisor Conjecture for the Heisenberg Group

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Excursions in Harmonic Analysis, Volume 3

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

Abstract

This chapter reports on the current status of the HRT Conjecture (also known as the linear independence of time–frequency shifts conjecture), and discusses its relationship with a longstanding conjecture in algebra known as the zero divisor conjecture.

∗ This work was partially supported by grants from the Simons Foundation (#229035 Heil; #244953 Speegle).

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References

  1. Balan R. The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time–frequency shift operators. Trans Am Math Soc. 2008;360:3921–41.

    Article  MATH  MathSciNet  Google Scholar 

  2. Balan R, Krishtal I. An almost periodic noncommutative Wiener’s lemma. J Math Anal Appl. 2010;370:339–49.

    Article  MATH  MathSciNet  Google Scholar 

  3. Benedetto JJ, Bourouihiya A. Linear independence of finite Gabor systems determined by behavior at infinity. J Geom Anal.2014; to appear.

    Google Scholar 

  4. Bownik M, Speegle D. Linear independence of Parseval wavelets. Illinois J Math. 2010;54:771–85.

    MATH  MathSciNet  Google Scholar 

  5. Bownik M, Speegle D. Linear independence of time-frequency translates of functions with faster than exponential decay. Bull Lond Math Soc. 2013;45:554–66.

    Article  MATH  MathSciNet  Google Scholar 

  6. Christensen O, Lindner AM. Lower bounds for finite wavelet and Gabor systems. Approx Theory Appl (N.S.). 2001;17:18–29.

    MATH  MathSciNet  Google Scholar 

  7. Daubechies I. Ten lectures on wavelets. Philadelphia: SIAM;1992.

    Book  MATH  Google Scholar 

  8. Demeter C. Linear independence of time frequency translates for special configurations. Math Res Lett. 2010;17:761–79.

    Article  MATH  MathSciNet  Google Scholar 

  9. Demeter C, Gautam SZ. On the finite linear independence of lattice Gabor systems. Proc Am Math Soc. 2013;141:1735–47.

    Article  MATH  MathSciNet  Google Scholar 

  10. Demeter C, Zaharescu A. Proof of the HRT conjecture for \((2,2)\) configurations. J Math Anal Appl. 2012;388:151–9.

    Article  MATH  MathSciNet  Google Scholar 

  11. Formanek E. The zero divisor question for supersolvable groups. Bull Aust Math Soc. 1973;9:69–71.

    Article  MATH  MathSciNet  Google Scholar 

  12. Gröchenig K. Linear independence of time-frequency shifts? Preprint. 2014.

    Google Scholar 

  13. Haar A. Zur Theorie der orthogonalen Funktionensysteme. Math Ann. 1910;69:331–71; English translation in [17].

    Google Scholar 

  14. Heil C. Linear independence of finite Gabor systems, In: Heil C, Editor. Harmonic analysis and applications. Boston: Birkhäuser; 2006. p. 171–206.

    Chapter  Google Scholar 

  15. Heil C. A basis theory primer, expanded Edition. Boston: Birkhäuser; 2011.

    Book  Google Scholar 

  16. Heil C. WHAT IS … a frame? Notices Am Math Soc. 2013;60:748–50.

    Article  MathSciNet  Google Scholar 

  17. Heil C, Walnut DF, Editors. Fundamental papers in wavelet theory. Princeton: Princeton University Press; 2006.[AQ1]

    MATH  Google Scholar 

  18. Heil C, Ramanathan J, Topiwala P. Linear independence of time-frequency translates. Proc Am Math Soc. 1996;124:2787–95.

    Article  MATH  MathSciNet  Google Scholar 

  19. Higman G. The units of group-rings. Proc Lond Math Soc. 1940;46(2):231–48.

    Article  MathSciNet  Google Scholar 

  20. Kutyniok G. Linear independence of time–frequency shifts under a generalized Schrödinger representation. Arch Math. (Basel) 2002;78:135–44.

    Article  MATH  MathSciNet  Google Scholar 

  21. Linnell PA. Analytic versions of the zero divisor conjecture. In: Kropholler PH, Niblo GA, Stöhr R, Editors. Geometry and cohomology in group theory. London Mathematical Society Lecture Note Ser., Vol. 252. Cambridge: Cambridge University Press; 1998. p. 209–48.

    Google Scholar 

  22. Linnell PA. Von Neumann algebras and linear independence of translates. Proc Am Math Soc. 1999;127:3269–77.

    Article  MATH  MathSciNet  Google Scholar 

  23. Passman DS. The algebraic structure of group rings. New York: Wiley-Interscience; 1977.

    MATH  Google Scholar 

  24. Rosenblatt J. Linear independence of translations. Int J Pure Appl Math. 2008;45:463–73.

    MATH  MathSciNet  Google Scholar 

  25. Snider RL. The zero divisor conjecture. In: McDonald BR, Morris RA, Editors. Ring theory, II. Lecture notes in pure and applied mathematics. Vol. 26. New York: Dekker; 1977. p. 261–95.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 30332, Atlanta, GA, USA

    Christopher Heil

  2. Department of Mathematics, Saint Louis University St. Louis, 63103, Louis, MO, USA

    Darrin Speegle∗

Authors
  1. Christopher Heil
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  2. Darrin Speegle∗
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Corresponding author

Correspondence to Christopher Heil .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Radu Balan

  2. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Matthew J. Begué

  3. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, Norbert Weiner Center, College Park, Maryland, USA

    Wojciech Czaja

  5. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Kasso A. Okoudjou

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Heil, C., Speegle∗, D. (2015). The HRT Conjecture and the Zero Divisor Conjecture for the Heisenberg Group. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 3. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13230-3_7

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