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 $\mathbb{F}$. We don't assume that E is complete and therefore we present an argument avoiding Baire's theorem.

中文翻译：

---

树上的广义谐波函数：普遍性和频繁普遍性

最近，已经研究了树T上的谐波函数和通常的泛谐波函数，并在整个场上采用可分离的Fréchet空间E取值。$\mathbb{C}$ 或者 $\mathbb{[R}$。在本文中，我们允许函数在一个相当普通的字段上取向量空间E中的值$\mathbb{F}$。可分离拓扑向量空间E的度量是平移不变的，除谐波函数外，我们还可以研究由系数为1的线性组合定义的更一般的函数$\mathbb{F}$。我们不认为E是完整的，因此我们提出一个避免Baire定理的论点。