Complex Hadamard matrices and the spectral set conjecture

• Autores: Mihail N. Kolounzakis, Máté Matolcsi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. Extra 1, 2006 (Ejemplar dedicado a: Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations), págs. 281-291
• Idioma: español
• Enlaces
  • Texto completo
• Resumen

  • By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction ¿spectral ) tile¿ of the Spectral Set Conjecture, for all sets A of size |A|  5, in any finite Abelian group.

    This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd.

    It was pointed out recently in [16] that the corresponding statement fails for |A| = 6 in the group Z5 3, and this observation quickly led to the failure of the Spectral Set Conjecture in R5 [16], and subsequently in R4 [13]. In the second part of this note we reduce this dimension further, showing that the direction ¿spectral ) tile¿ of the Spectral Set Conjecture is false already in dimension 3.

• Referencias bibliográficas
  • P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A. 37 (2004), 5355–5374.
  • B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Analysis 16 (1974), 101–121.
  • M.R. Garey and D.S. Johnson, Computers and Intractability, W\.H. Freeman and Co., San Francisco, Calif., 1979.
  • U. Haagerup, Orthogonal maximal abelian \*-subalgebras of the $n \times n$ matrices and cyclic $n$-roots, in Operator algebras and quantum...
  • A. Iosevich, N.H. Katz, and T. Tao, Convex bodies with a point of curvature do not have Fourier bases, Amer. J. Math. 123 (2001), 115–120.
  • A. Iosevich, N.H. Katz, and T. Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (2003), 559–569.
  • M.N. Kolountzakis, Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), 542–550.
  • M.N. Kolountzakis, The study of translational tiling with Fourier analysis, in Fourier analysis and convexity, 131–187, Appl. Numer. Harmon....
  • M.N. Kolountzakis and M. Matolcsi, Tiles with no spectra, Forum Math., to appear.
  • S. Konyagin and I. Laba, Spectra of certain types of polynomials and tiling of integers with translates of finite sets, J. Number Theory 103...
  • J.C. Lagarias and S. Szabó, Universal spectra and Tijdeman’s conjecture on factorization of cyclic groups, J. Fourier Anal. Appl. 7 (2001),...
  • J.C. Lagarias and Y. Wang, Spectral sets and factorizations of finite abelian groups, J. Funct. Anal. 145 (1997), 73–98.
  • P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A. 37 (2004), 5355–5374.
  • B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Analysis 16 (1974), 101–121.
  • M.R. Garey and D.S. Johnson, Computers and Intractability, W\.H. Freeman and Co., San Francisco, Calif., 1979.
  • U. Haagerup, Orthogonal maximal abelian \*-subalgebras of the $n \times n$ matrices and cyclic $n$-roots, in Operator algebras and quantum...
  • A. Iosevich, N.H. Katz, and T. Tao, Convex bodies with a point of curvature do not have Fourier bases, Amer. J. Math. 123 (2001), 115–120.
  • A. Iosevich, N.H. Katz, and T. Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (2003), 559–569.
  • M.N. Kolountzakis, Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), 542–550.
  • M.N. Kolountzakis, The study of translational tiling with Fourier analysis, in Fourier analysis and convexity, 131–187, Appl. Numer. Harmon....
  • M.N. Kolountzakis and M. Matolcsi, Tiles with no spectra, Forum Math., to appear.
  • S. Konyagin and I. Laba, Spectra of certain types of polynomials and tiling of integers with translates of finite sets, J. Number Theory 103...
  • J.C. Lagarias and S. Szabó, Universal spectra and Tijdeman’s conjecture on factorization of cyclic groups, J. Fourier Anal. Appl. 7 (2001),...
  • J.C. Lagarias and Y. Wang, Spectral sets and factorizations of finite abelian groups, J. Funct. Anal. 145 (1997), 73–98.M. Matolcsi, Fuglede’s...
  • S. Pedersen and Y. Wang, Universal spectra, universal tiling sets and the spectral set conjecture, Math. Scand. 88 (2001), 246–256.
  • W. Rudin, Fourier Analysis on Groups, Reprint of the 1962 original, John Wiley and Sons, Inc., New York, 1990.
  • T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), 251–258.