A class of non-convex polytopes that admit no orthonormal basis of exponentials

@article{Kolountzakis2001ACO,
  title={A class of non-convex polytopes that admit no orthonormal basis of exponentials},
  author={Mihail N. Kolountzakis and Michael Papadimitrakis},
  journal={Illinois Journal of Mathematics},
  year={2001},
  volume={46},
  pages={1227-1232}
}
A conjecture of Fuglede states that a bounded measurable set ⊂ R d , of measure 1, can tile R d by translations if and only if the Hilbert space L 2 () has an orthonormal basis consisting of exponentials e�(x) = exp 2πih λ, xi . If has the latter property it is called spectral. Let be a polytope in R d with the following property: there is a direction ξ ∈ S d 1 such that, of all the polytope faces perpendicular to ξ, the total area of the faces pointing in the positive ξ direction is more than… Expand
Spectrality of product domains and Fuglede’s conjecture for convex polytopes
A set Ω ⊂ ℝ 2 is said to be spectral if the space L 2 (Ω) has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets “behave like” sets which can tile theExpand
Spectrality of Polytopes and Equidecomposability by Translations
Let $A$ be a polytope in ${\mathbb{R}}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. AExpand
Tiling and spectral properties of near-cubic domains
We prove that if a measurable domain tiles R or R 2 by translations, and if it is \close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similarExpand
Spectra of certain types of polynomials and tiling of integers with translates of finite sets
We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can beExpand
Fuglede's spectral set conjecture for convex polytopes
Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture,Expand
Fuglede’s conjecture fails in dimension 4
In this note we modify a recent example of Tao and give an example of a set Omega subset of R-4 such that L-2(Omega) admits an orthonormal basis of exponentials {1/vertical bar Omega vertical bar 1/2Expand
Fuglede's conjecture is false in 5 and higher dimensions
We give an example of a set Ω ⊂ R 5 which is a finite union of unit cubes, such that L 2 (Ω) admits an orthonormal basis of exponentials { 1 |Ω|1/2 e 2πiξj ·x :
Non-symmetric convex polytopes and Gabor orthonormal bases
In this paper, we show that non-symmetric convex polytopes cannot serve as a window function to produce a Gabor orthonormal basis by any time-frequency sets.
Duality properties between spectra and tilings
Spectra and tilings play an important role in analysis and geometry respectively. The relations between spectra and tilings have baffled the mathematicians for a long time. Many conjectures, such asExpand
The Study of Translational Tiling with Fourier Analysis
In this survey I will try to describe how Fourier analysis is used in the study of translational tiling. Right away I will emphasize two restrictions that separate this area from the general theoryExpand
...
1
2
...

References

SHOWING 1-10 OF 10 REFERENCES
Non-symmetric convex domains have no basis of exponentials
A conjecture of Fuglede states that a bounded measurable set $\Omega$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(\Omega)$ has an orthonormal basisExpand
Commuting self-adjoint partial differential operators and a group theoretic problem
In Rn let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in L2(Ω)). The existence of n commuting self-adjoint operators H1,…, HnExpand
Convex bodies with a point of curvature do not have Fourier bases
We prove that no smooth symmetric convex body Ω with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The nonsymmetric case was proven in aExpand
The spectral set conjecture and multiplicative properties of roots of polynomials
Fuglede’s conjecture [2] states that a set Ω ⊂ Rn tiles R by translations if and only if L(Ω) has an orthogonal basis of exponentials. We obtain new partial results supporting the conjecture inExpand
Fuglede’s conjecture for a union of two intervals
We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations.
Convex Bodies: The Brunn-Minkowski Theory
1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.
Almost Periodic Functions
This new Dover edition is a reiseue of the first edition republished through epeci sl permission of Cambridge University Press.
Convex bodies with a point of curvature do not admit exponential bases, Amer
  • J. Math
  • 2001
Convex bodies with a point of curvature do not admit exponential bases
  • Amer. J. Math
Lb] I
  • Lb] I