Spectrum is periodic for n-intervals

Bose, Debashish; Madan, Shobha

doi:10.1016/j.jfa.2010.09.011

Cited by 16 publications

(25 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we study the case of spectral pairs (Ω, Λ) for bounded Borel subsets of R; a key ingredient is a result from [BM11,IK12] that the spectrum Λ has a finite period. But first we show that the spectral property implies the existence of a certain unitary group of local translations and we give a detailed description of this unitary group and various equivalent forms.…”

Section: General Spectral Subsets ω Of Rmentioning

confidence: 99%

See 1 more Smart Citation

Unitary groups and spectral sets

Dutkay

Jørgensen

2015

Journal of Functional Analysis

View full text Add to dashboard Cite

Abstract. We study spectral theory for bounded Borel subsets of R and in particular finite unions of intervals. For Hilbert space, we take L 2 of the union of the intervals. This yields a boundary value problem arising from the minimal operator D = 1 2πi d dx with domain consisting of C ∞ functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding self-adjoint extensions of D and for the associated unitary groups of local translations. While motivated by scattering theory and quantum graphs, our present focus is on the Fuglede-spectral pair problem. Stated more generally, this problem asks for a determination of those bounded Borel sets Ω in R k such that L 2 (Ω) has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to Ω. In the general case, we characterize Borel sets Ω having this spectral property in terms of a unitary representation of (R, +) acting by local translations.The case of k = 1 is of special interest, hence the interval-configurations. We give a characterization of those geometric interval-configurations which allow Fourier spectra directly in terms of the self-adjoint extensions of the minimal operator D. This allows for a direct and explicit interplay between geometry and spectra.

show abstract

Section: General Spectral Subsets ω Of Rmentioning

confidence: 99%

“…One of the main ingredients that we will use is the fact that any spectrum is periodic (see [BM11,IK12]). …”

Section: General Spectral Subsets ω Of Rmentioning

confidence: 99%

Unitary groups and spectral sets

Dutkay

Jørgensen

2015

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…with λ 0 = 0. Further, by the structure theorem proved in [3] we know that Λ is also a spectrum for a set Ω 1 which is a union of d equal intervals, whose end points lie on the lattice Z/d; i.e.,…”

mentioning

confidence: 99%

On the Rationality of the Spectrum

Bose

Madan

2017

J Fourier Anal Appl

Self Cite

View full text Add to dashboard Cite

Let Ω ⊂ R be a compact set with measure 1. If there exists a subset Λ ⊂ R such that the set of exponential functions E Λ := {e λ (x) = e 2πiλx | Ω : λ ∈ Λ} is an orthonormal basis for L 2 (Ω), then Λ is called a spectrum for the set Ω. A set Ω is said to tile R if there exists a set T such that Ω + T = R. A conjecture of Fuglede suggests that Spectra and Tiling sets are related. Lagarias and Wang [14] proved that Tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in[3], [8]. In this paper, we give some partial results to support the rationality of the spectrum.

show abstract

“…In this paper we present a new proof of the periodicity of the spectrum, which is a considerable simplification of that in [1]. [1]) If = n j =1 (a j , b j ) ⊆ R is a finite union of intervals of total length 1 and ⊆ R is a spectrum of then there exists a positive integer T such that + T = .…”

mentioning

confidence: 99%

“…[1]) If = n j =1 (a j , b j ) ⊆ R is a finite union of intervals of total length 1 and ⊆ R is a spectrum of then there exists a positive integer T such that + T = .…”

mentioning

confidence: 99%

Periodicity of the Spectrum of a Finite Union of Intervals

Kolountzakis

2011

J Fourier Anal Appl

View full text Add to dashboard Cite

A set , of Lebesgue measure 1, in the real line is called spectral if there is a set of real numbers such that the exponential functions e λ (x) = exp(2πiλx), λ ∈ , form a complete orthonormal system on L 2 ( ). Such a set is called a spectrum of . In this note we present a simplified proof of the fact that any spectrum of a set which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.

show abstract

Spectrum is periodic for n-intervals

Cited by 16 publications

References 49 publications

Unitary groups and spectral sets

Unitary groups and spectral sets

On the Rationality of the Spectrum

Periodicity of the Spectrum of a Finite Union of Intervals

Contact Info

Product

Resources

About