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Harmonic Analysis Invariants for Infinite Graphs Via Operators and Algorithms

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Abstract

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, probability, harmonic functions, and boundary theory. The limit constructions, finite to infinite, and local to global, can be used in various applications.

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Notes

  1. The terms electrical network, weighted network, electrical resistance network are used as synonyms in many papers and books on this subject.

  2. One can consider complex-valued functions in this (and other) definition; obvious changes can be easily made.

  3. It follows from Proposition 3.6 that the definition of a dipole given in Definition 2.2 agrees with that mentioned in (3.5)

  4. More generally, L is semibounded if for some number c one has \(\langle L f, f\rangle _{H} \ge c \langle f, f\rangle _{H}\).

  5. We deal with \(\Delta _{\mathcal {H}_E}\) here; the operator \(\Delta _2\) is considered similarly.

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Acknowledgements

The authors are pleased to thank our colleagues and collaborators. We are especially thankful to the members of the seminars in Mathematical Physics and Operator Theory at the University of Iowa for many helpful conversations. The authors express their gratitude to the referee for careful reading and useful suggestions.

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  1. Department of Mathematics, University of Iowa, Iowa City, IA, USA

    Sergey Bezuglyi & Palle E. T. Jorgensen

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  1. Sergey Bezuglyi
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Bezuglyi, S., Jorgensen, P.E.T. Harmonic Analysis Invariants for Infinite Graphs Via Operators and Algorithms. J Fourier Anal Appl 27, 34 (2021). https://doi.org/10.1007/s00041-021-09827-0

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