Generalized Harmonic Functions on Trees: Universality and Frequent Universality

TL;DR

This paper extends the study of harmonic and frequently universal functions on trees to more general vector spaces over arbitrary fields, removing the completeness assumption and introducing new methods.

Contribution

It generalizes previous results by allowing functions to take values in non-complete, topological vector spaces over general fields, and develops new proof techniques avoiding Baire's theorem.

Findings

Extended universality results to broader vector spaces

Developed new proof methods without Baire's theorem

Established universality properties for generalized functions

Abstract

Recently, harmonic functions and frequently universal harmonic functions on a tree $T$ have been studied, taking values on a separable Fr\'{e}chet 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 complete and therefore we present a new argument, avoiding Baire's theorem.