Polynomial approximation on the boundary and strictly inside |
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Authors: | N A Shirokov V Totik |
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Institution: | 1. LOMI, Fontanka 27, 191011, St. Petersburg, Russia 2. Bolyai Institute, Aradi v. tere 1, 6720, Szeged, Hungary 3. Department of Mathematics, University of South Florida, 33620, Tampa, Florida, USA
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Abstract: | We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point zo∈δK, then geometric convergence insideK is possible only for functions that are analytic at zo. We also consider the possibility of approximation rates of the form exp(?cn β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ n }. |
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