Band limited functions on quantum graphs

Pesenson, Isaac Z.

doi:10.1090/s0002-9939-05-07981-5

Cited by 12 publications

(7 citation statements)

References 15 publications

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“…It should be also mentioned that the methods of 2.2 were extended to metric(quantum) and combinatorial graphs [48], [110], [112], [115], [121], [123]. According to the spectral theory for such operators [11] there exists a direct integral of Hilbert spaces X = X(λ)dm(λ) and a unitary operator F from H onto X, which transforms the domains of…”

Section: Andmentioning

confidence: 99%

Geometric Space–Frequency Analysis on Manifolds

Feichtinger

Führ

Pesenson

2016

J Fourier Anal Appl

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This paper gives a survey of methods for the construction of spacefrequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections.After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley-Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley-Wiener frames in L 2 (M), for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in L 2 (M), with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on M.For compact Riemannian manifolds the theory extends to Lp and Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the socalled product property for eigenfunctions of certain operators and proves a cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay.Throughout the paper, the general theory is exemplified with the help of various concrete and relevant examples, such as the unit sphere and the Poincaré half plane.

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Section: Andmentioning

confidence: 99%

Geometric Space–Frequency Analysis on Manifolds

Feichtinger

Führ

Pesenson

2016

J Fourier Anal Appl

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“…Our sheaf-theoretic approach allows sufficient generality to treat sampling on non-Euclidean spaces. Others have studied sampling on non-Euclidean spaces, for instance general Hilbert spaces [22], Riemann surfaces [31], symmetric spaces [10], the hyperbolic plane [12], combinatorial graphs [26], and quantum graphs [23,24]. We show that sheaves provide unified sufficiency conditions for perfect reconstruction on abstract simplicial complexes, which encompass all of the above cases.…”

Section: Historical Contextmentioning

confidence: 89%

“…Additionally, the topology of the underlying space on which the differential equation is written impacts the process of reconstruction from samples. [23,24,27] The unifying power of sheaf theory means that all of the examples in this section can be treated in the same way, according to the following procedure:…”

Section: Examplesmentioning

confidence: 99%

A Sheaf-Theoretic Perspective on Sampling

Robinson

2015

Sampling Theory, a Renaissance

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“…Most recently Aziz et al established a method based on a wave packet propagation on quantum graphs that allows to distinguish between structures in complex networks [35] thanks to many well studied properties of the Laplacian (e.g. finite speed of propagation [36] as opposed to a discrete Laplacian [37,38]) and its spectra in quantum graphs [39][40][41][42][43][44][45]. The idea of determining the a shape of an object based on observable dynamics on it goes back to the work of Kac in 1966 [46] in which he asks whether it is possible to hear the shape of a drum.…”

Section: Introductionmentioning

confidence: 99%

Discovering hidden layers in quantum graphs

Gajewski,

Sienkiewicz,

Hołyst

2020

Preprint

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Finding hidden layers in complex networks is an important and a non-trivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multi-layer system exist and if so then what is their extent, i.e. how many unknown layers there are. We assume that all information we have is the time evolution of a wave propagation on a single layer of a network and show that it is indeed possible to uncover that which is hidden by merely observing the dynamics. We present evidence on both synthetic and real world networks that the frequency spectrum of the wave dynamics can express distinct features in the form of additional frequency peaks. These peaks exhibit dependence on the number of layers taking part in the propagation and thus allowing for the extraction of said number. In fact, with sufficient observation time, one can fully reconstruct the row-normalized adjacency matrix spectrum. We compare this approach to the work of Aziz et al. in which there has been established a wave packet signature method for discerning between various single layer graphs and we modify it for the purposes of multi-layer systems.

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Band limited functions on quantum graphs

Cited by 12 publications

References 15 publications

Geometric Space–Frequency Analysis on Manifolds

Geometric Space–Frequency Analysis on Manifolds

A Sheaf-Theoretic Perspective on Sampling

Discovering hidden layers in quantum graphs

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