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Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces arxiv.org/abs/2504.18543

Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces

We employ the perspective of the functional equation satisfied by the classical Fourier transform to derive the Helgason Fourier transform map $Ω^{l}(G/K,W)\longrightarrowΩ^{k}(G/K\times G/P,V[χ]):f\longmapsto \widehat{f}:G/K\times G/P\mapsto V[χ]:(x,b)\longmapsto\widehat{f}(x,b)$ (for $W-$valued differential forms $f\in Ω^{l}(G/K,W)$) as the $G-$ invariant vector bundle-valued differential form $\widehat{f}$ on the product space $G/K\times G/P$ whose image under the vector bundle-valued Poisson transform is the fibre convolution-integral $φ^{U^{σ,ν}}_{τ,l,k}* f$ on $G/K,$ where $φ^{U^{σ,ν}}_{τ,l,k}$ is the $W-$valued $τ-$spherical $l-$form on $G/K.$ Explicitly, we prove that $$\widehat{f}_{l,k,λ}(x,b)=({\bf C_{o}(λ)}^{-1}\circβ^{V}(λ))\circ(\int_{G/K}φ^{U^{σν},t}_{τ,l,k}\wedgeπ^{*}_{K}f)(x),$$ where $b\in G/P$ is a consequence of the boundary map $β^{V}(λ)$ and ${\bf C_{o}(λ)}$ is the vector bundle-valued Harish-Chandra $c-$function. The Fourier transform map $Ω^{l}(G/K,W)\longrightarrowΩ^{k}(G/K\times G/P,W)$ $:f\mapsto f^{\triangle}$ $:(b,x)\longmapsto f^{\triangle}(b,x)$ is then established to be explicitly given as $f^{\triangle}_{l,k,λ}(b,x)=$ $$\int_{G/P}ϕ_{k,l}\wedgeπ^{*}_{P}(({\bf C_{o}(λ)}^{-1}\circβ^{V}(λ))\circ(\int_{G/K}φ^{U^{σν},t}_{τ,l,k}\wedgeπ^{*}_{K}f)(x)).$$

arXiv.org

On the almost algebraicity of groups of automorphisms of connected Lie groups arxiv.org/abs/2504.18641

On the almost algebraicity of groups of automorphisms of connected Lie groups

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let Aut$(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of GL$(\frak G)$, where $\frak G$ is the Lie algebra of $G$. It is known (see Dani (1992) and Previts-Wu (2001)) that when $C$ is trivial Aut$(G)$ is almost algebraic, in the sense that it is open in an algebraic subgroup of GL$(\frak G)$, and in particular has only finitely many connected components. In this paper we analyse the situation further in this respect, with $C$ possibly nontrivial, and identify obstructions for almost algebraicity to hold; the criteria are in terms of the group of restrictions of automorphisms of $G$ to $C$, and the abelian quotient Lie group $G/\overline{[G,G]}C$ (see Theorem 1.1 for details). For the class of Lie groups which admit a finite-dimensional representation with discrete kernel (called class $\mathcal{C}$ groups) this yields a more precise description as to when Aut$(G)$ is almost algebraic (see Corollary 1.3), while in the general case a variety of patterns are observed (see §6). Along the way we also study almost algebraicity of subgroups of Aut$(G)$ fixing each point of a given torus in $G$, containing $C$ (see in particular Theorem 1.5), which also turns out to be of independent interest.

arXiv.org

Performance Analysis and Experimental Validation of UAV Corridor-Assisted Networks arxiv.org/abs/2504.18654 .IT

Performance Analysis and Experimental Validation of UAV Corridor-Assisted Networks

Unmanned aerial vehicle (UAV) corridor-assisted communication networks are expected to expand significantly in the upcoming years driven by several technological, regulatory, and societal trends. In this new type of networks, accurate and realistic channel models are essential for designing reliable, efficient, and secure communication systems. In this paper, an analytical framework is presented that is based on one-dimensional (1D) finite point processes, namely the binomial point process (BPP) and the finite homogeneous Poisson point process (HPPP), to model the spatial locations of UAV-Base Stations (UAV-BSs). To this end, the shadowing conditions experienced in the UAV-BS-to-ground users links are accurately considered in a realistic maximum power-based user association policy. Subsequently, coverage probability analysis under the two spatial models is conducted, and exact-form expressions are derived. In an attempt to reduce the analytical complexity of the derived expressions, a dominant interferer-based approach is also investigated. Finally, the main outcomes of this paper are extensively validated by empirical data collected in an air-to-ground measurement campaign. To the best of the authors' knowledge, this is the first work to experimentally verify a generic spatial model by jointly considering the random spatial and shadowing characteristics of a UAV-assisted air-to-ground network.

arXiv.org

The dimension spectrum of the infinitely generated Apollonian gasket arxiv.org/abs/2504.17835

Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials arxiv.org/abs/2504.17879

Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials

We study discrete-time Markov chains on countably infinite state spaces, which are perturbed by rather general confining (i.e.\ growing at infinity) potentials. Using a discrete-time analogue of the classical Feynman--Kac formula, we obtain two-sided estimates for the $n$-step heat kernels $u_n(x,y)$ of the perturbed chain. These estimates are of the form $u_n(x,y)\asymp λ_0^nϕ_0(x)\widehatϕ_0(y)+F_n(x,y)$, where $ϕ_0$ (and $\widehatϕ_0$) are the (dual) eigenfunctions for the lowest eigenvalue $λ_0$; the perturbation $F_n(x,y)$ is explicitly given, and it vanishes if either $x$ or $y$ is in a bounded set. The key assumptions are that the chain is uniformly lazy and that the \enquote{direct step property} (DSP) is satisfied. This means that the chain is more likely to move from state $x$ to state $y$ in a single step rather than in two or more steps. Starting from the form of the heat kernel estimate, we define the intrinsic (or ground-state transformed) chains and we introduce time-dependent ultracontractivity notions -- asymptotic and progressive intrinsic ultracontractivity -- which we can link to the growth behaviour of the confining potential; this allows us to consider arbitrarily slow growing potentials. These new notions of ultracontractivity also lead to a characterization of uniform (quasi-)ergodicity of the perturbed and the ground-state transformed Markov chains. At the end of the paper, we give various examples that illustrate how our findings relate to existing models, e.g.\ nearest-neighbour walks on infinite graphs, subordinate processes or non-reversible Markov chains.

arXiv.org

Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions arxiv.org/abs/2504.17899

Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.

arXiv.org
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