Concerning Exponential Bases on Multi-Rectangles of \(\boldsymbol {\mathbb {R}^d}\)

  • Laura De CarliEmail author
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We produce exponential bases with explicit frame constants on finite union of disjoint rectangles in \(\mathbb {R}^d\) with rational vertices.

2010Mathematics Subject Classification

42C15 42C30 


  1. 1.
    E. Agora, J. Antezana, C. Cabrelli, Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups. Adv. Math. 285, 454–477 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    N. Anderson, G. Best, A Gerschgorin-Rayleigh inequality for the eigenvalues of Hermitian matrices. Linear Multilinear Algebra 6(3), 219–222 (1978/1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Bezuglaya, Y. Katsnelson, The sampling theorem for functions with limited multi-band spectrum. Z. Anal. Anwendungen 12(3), 511–534 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Brualdi, S. Mellendorf, Regions in the complex plane containing the eigenvalues of a matrix. Am. Math. Mon. 101(10), 975–985 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    O. Christiansen, An Introduction to Frames and Riesz Bases (Birkhäuser, Boston, 2003)CrossRefGoogle Scholar
  6. 6.
    L. De Carli, A. Kumar, Exponential bases on two dimensional trapezoids. Proc. Am. Math. Soc. 143(7), 2893–2903 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. De Carli, G. Shaikh Samad, One-parameter groups of operators and discrete Hilbert transforms. Can. Math. Bull. 59, 497–507 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Dym, Linear Algebra in Action, Graduate Studies in Mathematics, vol. 78, 2nd edn. (American Mathematical Society, Providence, 2013)Google Scholar
  9. 9.
    B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Gerschgorin, Uber die abgrenzung der eigenwerte einer matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 6, 749–754 (1931)zbMATHGoogle Scholar
  11. 11.
    R. Gray, Toeplitz and circulant matrices: a review. Found. Trends Commun. Inf. Theory 2(3), 155–239 (2005)CrossRefGoogle Scholar
  12. 12.
    S. Grepstad, N. Lev, Multi-tiling and Riesz bases. Adv. Math. 252, 1–6 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    C. Heil, A basis theory primer, in Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, 2011)zbMATHGoogle Scholar
  14. 14.
    A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41, 367–379 (1936)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M.I. Kadec, The exact value of the Paley-Wiener constant. Sov. Math. Dokl. 5, 559–561 (1964)Google Scholar
  16. 16.
    M. Kolountzakis, The study of translational tiling with Fourier analysis, in Fourier Analysis and Convexity (Birkhäuser, Boston, 2004), pp. 131–187CrossRefGoogle Scholar
  17. 17.
    M. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials. Proc. Am. Math. Soc. 143, 741–747 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    G. Kozma, S. Nitzan, Combining Riesz bases. Invent. Math. 199(1), 267–285 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Kozma, S. Nitzan, Combining Riesz bases in Rd. Rev. Mat. Iberoam. 32(4), 1393–1406 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    I. Laba, Fuglede’s conjecture for a union of two interval. Proc. Am. Math. Soc. 129, 2965–2972 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    E. Laeng, Remarks on the Hilbert transform and some families of multiplier operators related to it. Collect. Math. 58(1), 25–44 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    J. Lagarias, J. Reeds, Y. Wang, Orthonormal bases of exponentials for the n-cube. Duke Math. J. 103(1), 25–37 (2000)MathSciNetCrossRefGoogle Scholar
  23. 23.
    N. Levinson, On non-harmonic Fourier series. Ann. Math. (2) 37(4), 919–936 (1936)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Y. Lyubarskii, K. Seip, Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets. J. Fourier Anal. Appl. 3(5), 598–615 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Marcus, H. Mint, A Survey of Matrix Theory and Matrix Inequalities (Prindle, Weber and Schmidt, Boston, 1964)Google Scholar
  26. 26.
    J. Marzo, Riesz basis of exponentials for a union of cubes in Rd.
  27. 27.
    B. Matei, Y. Meyer, A variant of compressed sensing. Rev. Mat. Iberoamericana 25(2), 669–692 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    S. Nitzan, A. Olevskii, A. Ulanovskii, Exponential frames for unbounded sets (2014). ArXiv:1410.5693Google Scholar
  29. 29.
    R. Paley, N. Wiener, in Fourier Transforms in the Complex Domain. American Mathematical Society Colloquium Publications, vol. 19 (American Mathematical Society, New York, 1934)Google Scholar
  30. 30.
    B. Pavlov, Basicity of an exponential system and Muckenhoupt’s condition. Sov. Math. Dokl. 20, 655–659 (1979)zbMATHGoogle Scholar
  31. 31.
    K. Seip, On the connection between exponential bases and certain related sequences in L2(−π, π). J. Funct. Anal. 130(1), 131–160 (1995)MathSciNetCrossRefGoogle Scholar
  32. 32.
    W. Sun, X. Zhou, On the stability of multivariate trigonometric systems. J. Math. Anal. Appl. 235, 159–167 (1999)MathSciNetCrossRefGoogle Scholar
  33. 33.
    R.M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980)zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFlorida International UniversityMiamiUSA

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