Geometry of spectral pairs

Bohnstengel, Jana; Jørgensen, Palle E. T.

doi:10.1007/s13324-011-0005-2

Cited by 3 publications

(3 citation statements)

References 32 publications

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“…(5) We briefly talk about the connection of the extension problem with the Fuglede conjecture. (6) We state the relation of the extension problem for positive definite functions with the extension problem for stationary Gaussian stochastic processes. (7) We outline the extension of our main result (theorem 1.6) to the class of conditionally negative definite functions, which has applications to harmonic analysis of Gaussian stochastic processes, whose increments in mean-square are stationary.…”

Section: 2mentioning

confidence: 99%

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Extension of positive definite functions

Niedzialomski

2015

Journal of Mathematical Analysis and Applications

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Let Ω ⊂ Rn be an open and connected subset of R n and letwhere Ω − Ω = {x − y : x, y ∈ Ω}, be a continuous positive definite function. We give necessary and sufficient conditions for F to have an extension to a continuous and positive definite function defined on the entire Euclidean space R n . The conditions are formulated in terms of strong commutativity of a system of certain unbounded selfadjoint operators defined on a Hilbert space associated to our function.Abstract Approved: Thesis Supervisor

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Section: 2mentioning

confidence: 99%

“…For background references on positive definite functions and their applications, see [5], [10], [25], [8], [6], [29], [28], [31]. In a special case, in [10], the idea of connecting spectral theory with the study of commuting selfadjoint extensions was suggested there.…”

Section: Introductionmentioning

confidence: 99%

Extension of positive definite functions

Niedzialomski

2015

Journal of Mathematical Analysis and Applications

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“…Our analysis of Gaussian processes associated to covariance kernels with singular measure σ is motivated in turn by a renewed interest in a harmonic analysis of Fourier decompositions in L 2 (σ) for the case when σ is singular and arises from a scale of selfsimilarities; see for example [21,10,14,23]. In order to obtain a more versatile harmonic analysis in the study factorizations, one is naturally led to consideration of independence, but for a host of problems [13,22], rather than the traditional notion of independence, one needs a related but different notion, that of free independence.…”

Section: Introductionmentioning

confidence: 99%

On free stochastic processes and their derivatives

Alpay

Jørgensen

Salomon

2014

Stochastic Processes and their Applications

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Abstract. We study a family of free stochastic processes whose covariance kernels K may be derived as a transform of a tempered measure σ. These processes arise, for example, in consideration non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal bases in the corresponding noncommutative L 2 of sample-space. We define a stochastic integral for our family of free processes.

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Unitary groups and spectral sets

Dutkay

Jørgensen

2015

Journal of Functional Analysis

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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.

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Geometry of spectral pairs

Abstract: In this paper we develop a geometric framework for spectral pairs and for orthogonal families of complex exponentials in L 2 (μ), where μ is a given Borel probability measure supported in R d .

Cited by 3 publications

References 32 publications

Extension of positive definite functions

Extension of positive definite functions

On free stochastic processes and their derivatives

Unitary groups and spectral sets

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