Algebraic genericity of frequently universal harmonic functions on trees

Double algebraic genericity of universal harmonic functions on trees

It is well known that the set of universal functions on a tree contains a vector space except zero which is dense in the set of harmonic functions. In this paper we improve this result by proving that the set of universal functions on a tree contains two vector spaces except zero which are dense in the space of harmonic functions and intersect only at zero.

Double algebraic genericity of universal harmonic functions on trees in the general case

It has been shown that the set of universal functions on trees contains a linear subspace except zero, dense in the space of harmonic functions. In this paper we show that the set of universal functions contains two linear subspaces except zero, dense in the space of harmonic functions that intersect only at zero. We work in the most general case that has been studied so far, letting our functions take values over a topological vector space.

拓扑泛性的一些结果

Some results of topological genericity

本文证明了在Hardy空间H^p（p<1）交集、A(D)及调和Hardy空间h^p（p<1）交集等函数空间中，满足Taylor系数无界或不属于ℓ^1等条件的函数集合具有拓扑泛性。

Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator

We solve a problem posed by Blasco, Bonilla and Grosse-Erdmann in 2010 by constructing a harmonic function on $\mathbb{R}^N$, that is frequently hypercyclic with respect to the partial differentiation operator $\partial/\partial x_k$ and which has a minimal growth rate in terms of the average $L^2$-norm on spheres of radius $r>0$ as $r \to \infty$.

Frequently dense harmonic functions and universal martingales on trees

We prove the existence of harmonic functions $f$ on trees, with respect to suitable transient transition operators $P$, that satisfy an analogue of Menshov universal property in the following sense: $f$ is the Poisson transform of a martingale on the boundary of the tree (equipped with the harmonic measure $m$ induced by $P$) such that, for every measurable function $h$ on the boundary, it contains a subsequence that converges to $h$ in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.

Generalized Harmonic Functions on Trees: Universality and Frequent Universality

Recently, harmonic functions and frequently universal harmonic functions on a tree $T$ have been studied, taking values on a separable Fréchet space $E$ over the field $\mathbb{C}$ or $\mathbb{R}$. In the present paper, we allow the functions to take values in a vector space $E$ over a rather general field $\mathbb{F}$. The metric of the separable topological vector space $E$ is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in $\mathbb{F}$. Unlike the past literature, we don't assume that $E$ is…