English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

On a semi-homogeneous tree, we study the $\ell^p$-spectrum of the Laplaceoperator $\mu_1$ (the isotropic nearest-neighbor transition operator); theknown results in the much simpler setting of homogeneous trees are obtained asparticular cases. The spectrum is given by the eigenvalues of sphericalfunctions, i.e., eigenfunctions of $mu_1$ that are radial with respect to areference vertex $v_0$ and normalized there. We show that spherical functionsare boundary integrals of generalized Poisson kernels that, unlike thehomogeneous setting, are not complex powers of the usual Poisson kernel. Wecompute these generalized Poisson kernels via Markov chains and theirgenerating functions, whence we work out explicit expressions for sphericalfunctions. On semi-homogeneous trees, spherical functions turn out to have an$\ell^p$ behavior that does not occur on homogeneous trees: one of them, for anappropriate choice of $v_0$, belongs to $\ell^p$ for some $p<2$.  Up to normalization, the operator $\mu_1^2$ differs from the step-2 Laplacian$\mu_2$ only by a shift. On the other hand, the recurrence relation associatedto the semi-homogeneous $\mu_2$ is that of a polygonal graph, akin to that ofthe Laplacian in a homogeneous tree. By this token, we compute the spectra of$\mu_1^2$ on a semi-homogeneous tree, hence, by extracting square roots, the$\ell^p$-spectrum of $\mu_1$ for $1\leqslant p <\infty$, and show that it isdisconnected for $p$ in an interval containing 2 but it is connected at allother values of $p$.

Spherical functions and spectrum of the Laplacian on semi-homogeneous trees