Related papers: Polyharmonic functions for finite graphs and Marko…
This paper studies the boundary behaviour of $\lambda$-polyharmonic functions for the simple random walk operator on a regular tree, where $\lambda$ is complex and $|\lambda|> \rho$, the $\ell^2$-spectral radius of the random walk. In…
We consider the open unit disk $\mathbb{D}$ equipped with the hyperbolic metric and the associated hyperbolic Laplacian $\mathfrak{L}$. For $\lambda \in \mathbb{C}$ and $n \in \mathbb{N}$, a $\lambda$-polyharmonic function of order $n$ is a…
On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with…
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
We study the map from conductances to edge energies for harmonic functions on finite graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of…
R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2)…
Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of pseudoharmonic function are obtained.
A 2p-times continuously differentiable complex valued function $f = u + iv$ in a simply connected domain is polyharmonic (or p-harmonic) if it satisfies the polyharmonic equation $\Delta^pF = 0$ . Every polyharmonic mapping f can be written…
We study the problem of approximating plurisubharmonic functions on a bounded domain $\Omega$ by continuous plurisubharmonic functions defined on neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to the problem of…
While discrete harmonic functions have been objects of interest for quite some time, this is not the case for discrete polyharmonic functions, as appear for instance in the asymptotics of path counting problems. In this article, a novel…
We study semifinite harmonic functions on the zigzag graph, which corresponds to Pieri's rule for the fundamental quasisymmetric functions $\{F_{\lambda}\}$. The main problem, which we solve here, is to classify the indecomposable…
We introduce a multivariate Markov transform which generalizes the well-known one-dimensional Stieltjes transform from the Moment problem and Spectral theory. Our main result states that two measures {\mu} and {\nu} with bounded support…
Polyharmonic functions f of infinite order and type {\tau} on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of infinite order and type…
First we extend the theory of subharmonic functions on smooth strictly $k$-analytic curves from Thuillier's thesis to the case of possibly singular analytic curves over a non-archimedean field. Classically psh functions are then defined as…
Assume that $f$ is Dunkl polyharmonic in $\mathbb{R}^n$ (i.e. $(\Delta_h)^p f=0$ for some integer $p$, where $\Delta_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $\kappa$, defined on $R$ and…
We extend holomorphically polyharmonic functions on a real ball to a complex set being the union of rotated balls. We solve a Dirichlet type problem for complex polyharmonic functions with the boundary condition given on the union of…
The study on the partial differential equations (systems) in the graph setting is a hot topic in recent years because of their applications to image processing and data clustering. Our motivation is to develop some existence results for…
We consider a countable tree $T$, possibly having vertices with infinite degree, and an arbitrary stochastic nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with…
We study semifinite harmonic functions on arbitrary branching graphs. We give a detailed exposition of an algebraic method which allows one to classify semifinite indecomposable harmonic functions on some multiplicative branching graphs.…
The paper is devoted to the study of compositions of polyharmonic mappings in simply connected domains. More precisely, we determine necessary and sufficient conditions of polyharmonic mapping $f$ such that $f\circ F$ (resp. $F\circ f$) is…