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## Some properties of spectral measures (2006)

Venue: | Appl. Comput. Harmon. Anal |

Citations: | 7 - 0 self |

### Citations

283 |
Entire functions
- Boas
- 1954
(Show Context)
Citation Context ...Λ is contained in the set Zp := {ξ ∈ R : p(ξ) = 0} ∪ {0}.8 I. LABA AND Y. WANG Since p is an entire function of exponential type with the diameter of suppp = Ω being less than a, it is well known =-=[Boa54]-=- that 3 2 D+(Zp) ≤ diam (Ω) < 3 2 a. This yields D+(Λ − Λ) < 3 2a. Proposition 2.2 now implies that Λ − Λ is a group. Since Λ − Λ is discrete and the only discrete subgroups of R are cyclic groups, Λ ... |

192 |
Necessary Density Conditions for Sampling and Interpolation of Certain Entire Functions
- Landau
- 1967
(Show Context)
Citation Context ...ry f ∈ L2 (Ω) we have ∑ |〈f, eλ〉| 2 ≥ C‖f‖ 2 2. λ∈Λ We say that Λ ⊆ Rd is an interpolation of B(Ω) if for every {cλ} ∈ l2 (Λ) there exists an f ∈ L2 (Ω) such that cλ = 〈f, eλ〉. A theorem of H. Landau =-=[Lan67]-=- states that D−(Λ) ≥ m(Ω) for a sampling Λ of B(Ω), and D+(Λ) ≤ m(Ω) for an interpolation Λ of B(Ω). Proof of Theorem 1.2. We first prove part (2). Let µ = ∑ a∈A paδa where A is a finite or countable ... |

114 |
Commuting self-adjoint partial differential operators and a group theoretic problem
- Fuglede
- 1974
(Show Context)
Citation Context ...o Ω. We say Ω is a spectral set if µ is a spectral measure. The main interest for studying spectral sets comes from its mysterious connection to tiling, first formulated in a conjecture by B. Fuglede =-=[Fug74]-=-, known today as the Fuglede Conjecture or the Spectral Set Conjecture. The Fuglede Conjecture. A measurable set Ω in R d is a spectral set if and only if it tiles R d by translation. The conjecture h... |

41 | Spectral sets and factorization of finite abelian groups
- Lagarias, Wang
- 1997
(Show Context)
Citation Context ...y evident, especially in low dimensions, as indicated by earlier works, including the original work of Fuglede [Fug74]. Many positive results have been established as well, see e.g. Lagarias and Wang =-=[LaWa97b]-=-, Pedersen and Wang [PeWa01], Laba [Lab01], and Iosevich, Katz and Tao [IKT03]. Interestingly, there is even evidence showing a strong connection between tiling and spectral measures, see [LabWa02] a... |

35 | Spectral pairs in cartesian coordinates - Jorgensen, Pedersen - 1999 |

34 | Fuglede’s conjecture is false in 5 and higher dimensions,” arXiv preprint math/0306134
- Tao
- 2003
(Show Context)
Citation Context ...A measurable set Ω in R d is a spectral set if and only if it tiles R d by translation. The conjecture had baffled the mathematicians who studied spectral sets for years until very recently, when Tao =-=[Tao04]-=- exhibited a spectral set in dimensions d ≥ 5 that is not a tile, and Kolountzakis and Matolsci [KoMa04] exhibited tiles that are not spectral sets in dimensions d ≥ 5. Despite the counterexamples, th... |

28 | Universal spectra, universal tiling sets and the spectral set conjecture
- Pedersen, Wang
(Show Context)
Citation Context ...dimensions, as indicated by earlier works, including the original work of Fuglede [Fug74]. Many positive results have been established as well, see e.g. Lagarias and Wang [LaWa97b], Pedersen and Wang =-=[PeWa01]-=-, Laba [Lab01], and Iosevich, Katz and Tao [IKT03]. Interestingly, there is even evidence showing a strong connection between tiling and spectral measures, see [LabWa02] and Strichartz [Str00]. Date:... |

27 | Non-symmetric convex domains have no basis of exponentials - Kolountzakis |

27 | The spectral set conjecture and multiplicative properties of roots of polynomials - Laba - 2002 |

26 | On spectral Cantor measures
- Laba, Wang
(Show Context)
Citation Context ...an orthogonal basis for L 2 (µ). In this case we call Λ a spectrum of µ, and (µ, Λ) a spectral pair. It should be pointed out that a measure µ may have more than one spectrum, see e.g. Laba and Wang =-=[LabWa02]-=-. Let Ω be a measurable set in R d and µ = m|Ω, the restriction of the d-dimensional Lebesgue measure m to Ω. We say Ω is a spectral set if µ is a spectral measure. The main interest for studying spec... |

24 | Orthonormal bases of exponentials for the n-cubes - Lagarias, Reeds, et al. |

24 |
Mock Fourier series and transforms associated with certain Cantor measures
- Strichartz
(Show Context)
Citation Context ... Wang [PeWa01], Laba [Lab01], and Iosevich, Katz and Tao [IKT03]. Interestingly, there is even evidence showing a strong connection between tiling and spectral measures, see [LabWa02] and Strichartz =-=[Str00]-=-. Date: October 21, 2004. The first author is partially supported by the Natural Science and Engineering Research Council of Canada. The second author is supported in part by the National Science Fund... |

23 |
Fuglede’s conjecture for a union of two intervals
- ̷Laba
(Show Context)
Citation Context ...ndicated by earlier works, including the original work of Fuglede [Fug74]. Many positive results have been established as well, see e.g. Lagarias and Wang [LaWa97b], Pedersen and Wang [PeWa01], Laba =-=[Lab01]-=-, and Iosevich, Katz and Tao [IKT03]. Interestingly, there is even evidence showing a strong connection between tiling and spectral measures, see [LabWa02] and Strichartz [Str00]. Date: October 21, 20... |

19 |
Dense analytic subspaces in fractal L 2-spaces
- Jorgensen, Pedersen
- 1998
(Show Context)
Citation Context ...In this paper we prove some properties of spectral measures. In particular, we prove results that highlight the 3/2-rule. 1. Introduction Spectral measures, first introduced by Jorgensen and Pedersen =-=[JoPe98]-=-, are a natural extension of spectral sets. Let µ be a finite Borel measure in R d . We say µ is a spectral measure if there exists a set Λ ⊂ R d such that the set of exponentials {exp(2πiλ · x) : λ ∈... |

12 | Universal spectra and Tijdeman’s conjecture on factorization of cyclic groups - Lagarias, Szabó |

8 | Fuglede conjecture holds for convex planar domains
- Iosevich, Katz, et al.
(Show Context)
Citation Context ... the original work of Fuglede [Fug74]. Many positive results have been established as well, see e.g. Lagarias and Wang [LaWa97b], Pedersen and Wang [PeWa01], Laba [Lab01], and Iosevich, Katz and Tao =-=[IKT03]-=-. Interestingly, there is even evidence showing a strong connection between tiling and spectral measures, see [LabWa02] and Strichartz [Str00]. Date: October 21, 2004. The first author is partially su... |

8 | Remarks on: “Dense analytic subspaces in fractal L2spaces - Strichartz - 1998 |

1 |
Tiles with no spectra, preprint arXiv:math.CA/0406127
- Kolountzakis, Matolcsi
(Show Context)
Citation Context ...had baffled the mathematicians who studied spectral sets for years until very recently, when Tao [Tao04] exhibited a spectral set in dimensions d ≥ 5 that is not a tile, and Kolountzakis and Matolsci =-=[KoMa04]-=- exhibited tiles that are not spectral sets in dimensions d ≥ 5. Despite the counterexamples, the connection between spectral sets and tiling is strongly evident, especially in low dimensions, as indi... |