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.

Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.