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

2021/08/29 by Khodr Shamseddine, Shamseddine, Khodr
Economics, Econometrics and Finance · Mathematics · #12J25 #26E20 #32P05 #41A58 #46S10 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Stochastic processes and financial applications #advanced mathematical theories #math.FA #msc:12J25 #msc:26E20 #msc:32P05 #msc:41A58 #msc:46S10

paper · pdf · doi:10.48550/arxiv.2108.12933

openalex publication_date 2021/08/29 · arxiv created 2021/09/11 · arxiv updated 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let 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^∞ functions at a point or on an open subset of N. Then we show under what conditions a WLUD^∞ function at a point x₀\inN is analytic in an interval around x₀, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD^k and WLUD^∞ to functions from Nⁿ to N, for any n∈ℕ. Then we formulate conditions under which a WLUD^∞ function at a point \boldsymbolx₀ ∈ Nⁿ is analytic at that point.

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