Orthogonal and Lq-extremal polynomials on inverse images of polynomial mappings

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 Pn on K with respect to the Lq(μ)-norm, q∈[1,∞), then zero is also a best approximation to f∘T on T−1(K) with respect to the Lq(μT)-norm, where μT arises from μ by the transformation T. In particular, μT 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 Lq(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.