Approximation by antiderivatives |
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Authors: | Wolfgang Luh |
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Institution: | 1. Fachbereich 4/Mathematik, Universit?t Trier, Postfach 38 25, D-5500, Trier, Germany
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Abstract: | Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ (?n)} ofn-fold antiderivatives (i.e., we haveφ (0)(z)∶=φ(z) andφ (?n)(z)=dφ (?n?1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold: - For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n k} such that {φ?nk} converges tof uniformly onB.
- For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m k} such that {φ?mk} converges tof compactly onU.
- For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p k} such that {φ (-Pk)} converges tof almost everywhere onE.
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