Fractional non-Archimedean calculus in one variable

Abstract

Let r ≥ 0 be a real number. We will introduce a notion of r-fold differentiability for functions in many variables over a non-Archimedeanly valued complete field K and then examine properties of theirs such as localness, completeness as a locally convex K-algebra, density of (locally) polynomial functions, closure under composition and, for the dual, under convolution. The definition of a Cr-function will be given through partial difference quotients and build up on the one-variable case already studied in [8]. In line with [2], we will also show a function on ℤpd to be r-times differentiable if and only if its Mahler coefficients obey |an‖n|r → 0 as | n| → ∞. As a corollary, a characterization of Cr-functions f: X → K on open X ⊆ ℚpd by partial Taylor-polynomials is obtained.