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A theorem of Roe and Strichartz on homogeneous trees

A theorem of Roe and Strichartz on homogeneous trees AbstractLet 𝔛{\mathfrak{X}} be a homogeneous tree and let ℒ{\mathcal{L}} be the Laplace operator on 𝔛{\mathfrak{X}}. In this paper, we address problems of the following form: Suppose that {fk}k∈ℤ{\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛{\mathfrak{X}} such that for all k∈ℤ{k\in\mathbb{Z}} one has ℒ⁢fk=A⁢fk+1{\mathcal{L}f_{k}=Af_{k+1}} and ∥fk∥≤M{\lVert f_{k}\rVert\leq M} for some constants A∈ℂ{A\in\mathbb{C}}, M>0{M>0} and a suitable norm ∥⋅∥{\lVert\,\cdot\,\rVert}. From this hypothesis, we try to infer that f0{f_{0}}, and hence every fk{f_{k}}, is an eigenfunction of ℒ{\mathcal{L}}. Moreover, we express f0{f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛{\mathfrak{X}}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

A theorem of Roe and Strichartz on homogeneous trees

Forum Mathematicum , Volume 34 (1): 22 – Jan 1, 2022

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Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2020-0358
Publisher site
See Article on Publisher Site

Abstract

AbstractLet 𝔛{\mathfrak{X}} be a homogeneous tree and let ℒ{\mathcal{L}} be the Laplace operator on 𝔛{\mathfrak{X}}. In this paper, we address problems of the following form: Suppose that {fk}k∈ℤ{\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛{\mathfrak{X}} such that for all k∈ℤ{k\in\mathbb{Z}} one has ℒ⁢fk=A⁢fk+1{\mathcal{L}f_{k}=Af_{k+1}} and ∥fk∥≤M{\lVert f_{k}\rVert\leq M} for some constants A∈ℂ{A\in\mathbb{C}}, M>0{M>0} and a suitable norm ∥⋅∥{\lVert\,\cdot\,\rVert}. From this hypothesis, we try to infer that f0{f_{0}}, and hence every fk{f_{k}}, is an eigenfunction of ℒ{\mathcal{L}}. Moreover, we express f0{f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛{\mathfrak{X}}.

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 2022

Keywords: Homogeneous tree; spectrum of Laplacian; eigenfunction of Laplacian; Fourier analysis; 43A85; 39A12; 20E08

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