2020/10/22 by Sergey Bezuglyi, Bezuglyi, Sergey, Palle E. T. Jørgensen +1
Computer Science · Mathematics · #05C60 #37B10 #41A63 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Graph theory and applications #Spectral Theory in Mathematical Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2010.12442
openalex publication_date 2020/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present recent advances in harmonic analysis on infinite graphs. Our\napproach combines combinatorial tools with new results from the theory of\nunbounded Hermitian operators in Hilbert space, geometry, boundary\nconstructions, and spectral invariants. We focus on particular classes of\ninfinite graphs, including such weighted graphs which arise in electrical\nnetwork models, as well as new diagrammatic graph representations. We further\nstress some direct parallels between our present analysis on infinite graphs,\non the one hand, and, on the other, specific areas of potential theory, Fourier\nduality, probability, harmonic functions, sampling/interpolation, and boundary\ntheory. With the use of limit constructions, finite to infinite, and local to\nglobal, we outline how our results for infinite graphs may be viewed as\nextensions of Shannon's theory: Starting with a countable infinite graph G,\nand a suitable fixed positive weight function, we show that there are certain\ncontinua (certain ambient sets X) extending G, and associated notions of\ninterpolation for (Hilbert spaces of) functions on X from their restrictions\nto the discrete graph G.\n