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Three Problems on Exponential Bases

  • Laura De Carli (a1), Alberto Mizrahi (a1) and Alexander Tepper (a1)

We consider three special and significant cases of the following problem. Let $D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let $E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of $\mathbb{R}^{d}$ . Find conditions on $D$ for which $E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for $L^{2}(D)$ .

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[1] Ajtai, M., Generating hard instances of lattice problems (extended abstract) . In: Proceedings of the twenty-eighth annual ACM symposium on the theory of computing (Philadelphia, PA, 1996), ACM, New York, 1996, pp. 99108.
[2] Aggarwal, D. and Dubey, C., Improved hardness results for unique shortest vector problem . Inform. Process. Lett. 116(2016), no. 10, 631637.
[3] Aldroubi, A., Sun, Q., and Tang, W., Connection between p–frames and p–Riesz bases in locally finite SIS of L p (ℝ). Proceedings of SPIE: The International Society for Optical Engineering, February 1970.
[4] Aldroubi, A., Sun, Q., and Tang, W., p-Frames and shift invariant subspaces of L p . J. Fourier Anal. Appl. 7(2001), 121.
[5] Agora, E., Antezana, J., and Cabrelli, C., Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups . Adv. Math. 285(2015), 454477.
[6] Selvan, A. and Radha, R., Sampling and reconstruction in shift invariant spaces on ℝ d . Ann. Mat. Pura Appl. 194(2015), no. 6, 16831706.
[7] Barbieri, D., Hernandez, E., and Mayeli, A., Lattice sub-tilings and frames in LCA groups . C. R. Math. Acad. Sci. Paris 355(2017), no. 2, 193199.
[8] Beurling, A., The collected works of Arne Beurling. Vol. 2, Contemporary Mathematics, Birkhäuser Boston Inc., Boston, MA, 1989.
[9] Beurling, A., Local harmonic analysis with some applications to differential operator . In: Some Recent Advances in the Basic Sciences, Vol. 1 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1962–1964), Belfer Graduate School of Science, Yeshiva Univ., New York, 1966, pp. 109125.
[10] Casazza, P., Christensen, O., and Stoeva, D. T., Frame expansions in separable Banach spaces . J. Math. Anal. Appl. 307(2005), 710723.
[11] Christiansen, O., An introduction to frames and Riesz bases. Applied and numerical harmonic analysis. Birkhäuser Boston, Inc., Boston, MA, 2003.
[12] Christiansen, O., Deng, B., and Heil, C., Density of Gabor frames . Appl. Comput. Harmon. Anal. 7(1999), 292304.
[13] Christensen, O. and Stoeva, D. T., p-frames in separable Banach spaces . Adv. Comput. Math. 18(2003), 117126.
[14] De Carli, L. and Kumar, A., Exponential bases on two dimensional trapezoids . Proc. Amer. Math. Soc. 143(2015), no. 7, 28932903.
[15] Fuglede, B., Commuting self-adjoint partial differential operators and a group theoretic problem . J. Functional Analysis 16(1974), 101121.
[16] Gabardo, J.-P. and Li, Y.-Z., Density results for Gabor systems associated with periodic subsets of the real line . J. Approx. Theory 157(2009), 172192.
[17] Grepstad, S. and Lev, N., Multi-tiling and Riesz bases . Adv. Math. 252(2014), 16.
[18] Haase, M., Functional analysis: an elementary introduction. Graduate studies in Mathematics, 156, American Mathematical Society, Providence, RI, 2014.
[19] Heil, C., A basis theory primer. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2011.
[20] Khot, S., Hardness of approximating the shortest vector problem in lattices . J. ACM. 52(2005), no. 5, 789808.
[21] Kolountzakis, M., Multiple lattice tiles and Riesz bases of exponentials . Proc. Amer. Math. Soc. 143(2015), 741747.
[22] Kolountzakis, M., The study of translational tiling with Fourier analysis. Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 131187.
[23] Jia, R. Q. and Micchelli, C. A., Using the refinement equation for the construction of pre-wavelets II: power of two . In: Curves and surfaces (Chamonix-Mont-Blanc, 1990), Academic Press, Boston, MA, 1991, pp. 209246.
[24] Laba, I., Fuglede’s conjecture for a union of two intervals . Proc. Amer. Math. Soc. 129(2001), no. 10, 29652972.
[25] Landau, H. J., Necessary density conditions for sampling and interpolation of certain entire functions . Acta Math. 117(1967), 3752.
[26] Nitzan, S. and Olevskii, A., Revisiting Landau’s density theorems for Paley-Wiener spaces . C. R. Acad. Sci. Paris 350(2012), no. 9–10, 509512.
[27] Seip, K., On the connection between exponential bases and certain related sequences in L 2(-𝜋, 𝜋) . J. Funct. Anal. 130(1995), no. 1, 131160.
[28] Young, R. M., An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, 93, Academic Press, New York, 1980.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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