Abstract
LetC be an analytic Jordan curve, letG be the interior ofC, and letU (w) be (at least) continuous onC. Here the solution of the Dirichlet problemu(w) which coincides withU(w) onC is approximated by harmonic polynomials. These harmonic polynomialsF n F(w) are determined by interpolatingU(w) in a given point system. For sufficiently greatn we prove |u(w)−F n (w)|≤K·logn·E n in\(\bar G\), where\(E_n = \mathop {\max }\limits_{w \in C} \left| {U(w) - h_n (w)} \right|\) andh n (w) is the harmonic polynomial of degreen of best approximation toU(w) onC andK is a constant independent ofn.
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Menke, K. Lösung des Dirichlet-Problems bei Jordangebieten mit analytischem Rand durch Interpolation. Monatshefte für Mathematik 80, 297–306 (1975). https://doi.org/10.1007/BF01472577
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DOI: https://doi.org/10.1007/BF01472577