Abstract
Let T be a polynomial of degree N and let K be a compact set with C . First it is shown, if zero is a best approximation to f from P n on K with respect to the L q ( μ)-norm, q∈[1,∞), then zero is also a best approximation to f∘ T on T −1(K) with respect to the L q(μ T ) -norm, where μ T arises from μ by the transformation T . In particular, μ T is the equilibrium measure on T −1(K) , if μ is the equilibrium measure on K. For q=∞, i.e., the sup-norm, a corresponding result is presented. In this way, polynomials minimal on several intervals, on lemniscates, on equipotential lines of compact sets, etc. are obtained. Special attention is given to L q ( μ)-minimal polynomials on Julia sets. Next, based on asymptotic results of Widom, we show that the minimum deviation of polynomials orthogonal with respect to a positive measure on T −1(∂K) behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour, too. Some open problems are also given.
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