Harmonic analysis on graphs via Bratteli diagrams and path-space measures

Bezuglyi, Sergey; Jørgensen, Palle E. T.

doi:10.48550/arxiv.2007.15566

Cited by 1 publication

(2 citation statements)

References 30 publications

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“…The notion of Bratteli diagrams is widely used in operator algebras, dimension groups, the theory of dynamical systems, and other adjoint areas. The reader can find more information in [Bra72], [GPS95], [HPS92], [Dur10], [BJ15], [BK16], [BK20], [BJ20], [Put18].…”

Section: Combinatorial Diagrams and Electrical Networkmentioning

confidence: 99%

“…This entails multiple recent research collaborations. Our emphasis is on both new and recent results from collaborations between the present co-authors, as well as joint results by the second named author and Erin Pearse; see especially [BJ15,BJ19a,BJ19b,BJ20] and [JP10, JP11, JP13, JP14, JP16, JPT18a]. In the present paper, we aim to address readers and researchers from diverse neighboring areas, as well as from applied areas.…”

Section: Introductionmentioning

confidence: 98%

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Harmonic analysis invariants for infinite graphs via operators and algorithms

Bezuglyi¹,

Jørgensen²

2020

Preprint

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We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, Fourier duality, probability, harmonic functions, sampling/interpolation, and boundary theory. With the use of limit constructions, finite to infinite, and local to global, we outline how our results for infinite graphs may be viewed as extensions of Shannon's theory: Starting with a countable infinite graph G, and a suitable fixed positive weight function, we show that there are certain continua (certain ambient sets X) extending G, and associated notions of interpolation for (Hilbert spaces of) functions on X from their restrictions to the discrete graph G.

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