On the analyticity of WLUD$^\infty$ functions of one variable and WLUD$^\infty$ functions of several variables in a complete non-Archimedean valued field

TL;DR

This paper investigates the conditions under which WLUD$^ $ functions in a complete non-Archimedean valued field are analytic, extending the concepts to multiple variables and establishing criteria for convergence of Taylor series.

Contribution

It generalizes the concept of WLUD$^ $ functions to several variables and provides conditions for their analyticity in a non-Archimedean setting.

Findings

WLUD$^ $ functions can be analytic under certain conditions.

Extension of WLUD$^ $ concepts to multivariable functions.

Criteria for Taylor series convergence in non-Archimedean fields.

Abstract

Let $\mathcal{N}$ be a non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^k$) functions, and WLUD$^\infty$ functions at a point or on an open subset of $\mathcal{N}$. Then we show under what conditions a WLUD$^\infty$ function at a point $x_0\in\mathcal{N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^k$ and WLUD$^\infty$ to functions from $\mathcal{N}^n$ to $\mathcal{N}$, for any $n\in\mathbb{N}$. Then we formulate conditions under which a WLUD$^\infty$ function at a point $\boldsymbol{x_0} \in \mathcal{N}^n$ is analytic at that point.