Riesz means on homogeneous trees

Effie Papageorgiou

doi:10.1515/conop-2020-0111

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-021-09884-5 ◽

2021 ◽

Vol 27 (5) ◽

Author(s):

Francesca Bartolucci ◽

Filippo De Mari ◽

Matteo Monti

Keyword(s):

Radon Transform ◽

Homogeneous Tree ◽

Homogeneous Trees ◽

Regular Representations ◽

Group Of Isometries

AbstractFollowing previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

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A theorem of Roe and Strichartz on homogeneous trees

Forum Mathematicum ◽

10.1515/forum-2020-0358 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Sumit Kumar Rano

Keyword(s):

Laplace Operator ◽

Infinite Sequence ◽

Homogeneous Tree ◽

Poisson Transform ◽

Formula One ◽

Homogeneous Trees ◽

Suitable Norm ◽

The Laplace Operator ◽

Sequence Of Functions

Abstract Let 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 { f k } 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 ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ 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 f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

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SCHUR MULTIPLIERS AND SPHERICAL FUNCTIONS ON HOMOGENEOUS TREES

International Journal of Mathematics ◽

10.1142/s0129167x10006537 ◽

2010 ◽

Vol 21 (10) ◽

pp. 1337-1382 ◽

Cited By ~ 6

Author(s):

U. HAAGERUP ◽

T. STEENSTRUP ◽

R. SZWARC

Keyword(s):

Fourier Multiplier ◽

Spherical Functions ◽

Sufficient Condition ◽

Homogeneous Tree ◽

Necessary And Sufficient Condition ◽

Closed Expression ◽

Radial Functions ◽

Schur Multipliers ◽

Homogeneous Trees ◽

Necessary And Sufficient

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ||ψ||S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ||⋅||M0A(G) of the radial functions on the free (non-abelian) group 𝔽N on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL2(ℚq) for every prime number q.

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Lp–Lrestimates for the Poisson semigroup on homogeneous trees

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700001269 ◽

1998 ◽

Vol 64 (1) ◽

pp. 20-32 ◽

Cited By ~ 2

Author(s):

Alberto G. Setti

Keyword(s):

Sharp Estimate ◽

Homogeneous Tree ◽

Homogeneous Trees ◽

Poisson Semigroup ◽

Operator Norms

AbstractLet be a homogeneous tree of degree at least three. In this paper we investigate for which values of p and r the (σθ)-Poisson semigroup is Lp – Lr,-bounded, and we sharp estimate for the corresponding operator norms.

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The range of the Helgason-Fourier transformation on homogeneous trees

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032858 ◽

1999 ◽

Vol 59 (2) ◽

pp. 237-246 ◽

Cited By ~ 7

Author(s):

Michael Cowling ◽

Alberto G. Setti

Keyword(s):

Reference Point ◽

Fourier Transformation ◽

Symmetric Spaces ◽

Closed Ball ◽

Homogeneous Tree ◽

Homogeneous Trees ◽

Fixed Reference ◽

Decreasing Functions

Let be a homogeneous tree, o be a fixed reference point in , and be the closed ball of radius N in centred at o. In this paper we characterise the image under the Helgason–Fourier transformation ℋ of , the space of functions supported in , and of , the space of rapidly decreasing functions on . In both cases our results are counterparts of known results for the Helgason–Fourier transformation on noncompact symmetric spaces.

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Measurable Events Indexed by Trees

Combinatorics Probability Computing ◽

10.1017/s0963548312000053 ◽

2012 ◽

Vol 21 (3) ◽

pp. 374-411 ◽

Cited By ~ 2

Author(s):

PANDELIS DODOS ◽

VASSILIS KANELLOPOULOS ◽

KONSTANTINOS TYROS

Keyword(s):

Finite Subset ◽

Probability Space ◽

Homogeneous Tree ◽

Homogeneous Trees ◽

Infinite Height ◽

Probability Spaces ◽

Image Position

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. We study the behaviour of measurable events in probability spaces indexed by homogeneous trees.Precisely, we show that for every integer b ≥ 2 and every integer n ≥ 1 there exists an integer q(b,n) with the following property. If T is a homogeneous tree with branching number b and {At:t ∈ T} is a family of measurable events in a probability space (Ω,Σ,μ) satisfying μ(At)≥ϵ>0 for every t ∈ T, then for every 0<θ<ϵ there exists a strong subtree S of T of infinite height, such that for every finite subset F of S of cardinality n ≥ 1 we have In fact, we can take q(b,n)= ((2b−1)2n−1−1)·(2b−2)−1. A finite version of this result is also obtained.

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Special series of unitary representations of groups acting on homogeneous trees

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700026513 ◽

1986 ◽

Vol 40 (1) ◽

pp. 83-88 ◽

Cited By ~ 3

Author(s):

Anna Maria Mantero ◽

Anna Zappa

Keyword(s):

Discrete Subgroup ◽

Unitary Representations ◽

Homogeneous Tree ◽

Special Series ◽

Representations Of Groups ◽

Homogeneous Trees

AbstractLet G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let be the space of functions on the Poission boundary ω, of zero mean on ω. When p is a prime. G is a discrete subgroup of PGL2(Qp) of finite covolume. The representations of the special series of PGL2(Qp), Which are irreducible and unitary in an appropriate completion of , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on and topologically irreducible on endowed with the week topology.

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Large Deviations for Radial Random Walks on Homogeneous Trees

Nagoya Mathematical Journal ◽

10.1017/s002776300002585x ◽

2007 ◽

Vol 187 ◽

pp. 75-90

Author(s):

Kanji Ichihara

Keyword(s):

Markov Chain ◽

Random Walk ◽

Large Deviations ◽

Random Walks ◽

Large Deviation ◽

Rate Function ◽

Homogeneous Tree ◽

Homogeneous Trees ◽

Radial Random Walks

AbstractDonsker-Varadhan’s type large deviation will be discussed for the pinned motion of a radial random walk on a homogeneous tree. We shall prove that the rate function corresponding to the large deviation is associated with a new Markov chain constructed from the above random walk through a harmonic transform based on a positive principal eigenfunction for the generator of the random walk.

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