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

摘要

Let J 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 o J on J~(-1) (K) with respect to the L_q(μ~J)-norm, where μ~J arises from μ by the transformation J. In particular, μ~J is the equilibrium measure on J~(-1) (K), if μ is the equilibrium measure on K. For q = ∞, i.e., the su-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 J~(-1) ((partial deriv)K) behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour, too. Some open problems are also given.