Fractional non-Archimedean calculus in one variable |
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Authors: | E. Nage |
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Affiliation: | 1. Institut de Math??matiques de Jussieu, 4 place Jussieu, 75005, Paris, France
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Abstract: | Let ?? be a natural number. A function f: ? p ?? K into a non-Archimedeanly valued complete field K ? ? p is ??-times continuously differentiable if and only if its Mahler coefficients (a n ) n??? obey |a n |n ?? ?? 0 as n ?? ??. For a real number r ?? 0, this suggests the ad hoc definition by [1] of a C r -function f: ? p ?? K by asking its Mahler coefficients (a n ) n??? to satisfy |a n |n r ?? 0 as n?? ??. We will present for functions f: X ?? K on subsets X ? K without isolated points a general pointwise notion of r-fold differentiability through iterated difference quotients, subsequently shown on the domain X = ? p to coincide with the one given above. For functions on open domains, we prove this notion to admit a handier characterization by its Taylor polynomial up to degree ?r?. |
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