Fractional non-Archimedean differentiability [Elektronische Ressource] / vorgelegt von Enno Nagel

Fractional Non-ArchimedeanDifferentiabilityEnno Nagel2011MathematikFractional Non-ArchimedeanDifferentiabilityInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakultätder Westfälischen Wilhelms-Universität Münstervorgelegt vonENNO NAGELaus Stadtlohn-2011-Dekan: PROFESSOR DR. MATTHIAS LÖWEErster Gutachter: PROFESSOR DR. PETER SCHNEIDERZweiter PROFESSOR DR. CHRISTOPHER DENINGERTag der mündlichen Prüfung: 01.02.2011Tag der Promotion:AbstractLetr≥ 0 be real number letK be a complete non-Archimedeanly non-trivially valued field.dIn the first chapter, we give the definition of a functionf : X→ E on a domainX⊆ Krwith values in aK-Banach spaceE to ber-times differentiable orC at a pointa∈X. Thenrwe endow the K-vector space of all suchC -functions with a locally convex topology andexamine properties of theirs such as completeness, density of (locally) polynomial functions,closure under composition and, for the dual, under convolution.For functions on open domains in one variable, we show this definition to equal a handier de-rscription through the convergence speedo(1/|h| ) of the rest term of the Taylor-polynomial atdx +h expanded aroundx up to degreebrc. Moreover on the special domainX =Z we showpr dtheC -functionsf :Z →K to be characterized by its Mahler coefficients (a ) d obeyingn n∈Npr|a ||n| → 0 as|n|→∞, where we put|n| :=n +···+n .