Neamtu, Marian (ed.) et al., Approximation theory XIII: San Antonio 2010. Selected papers based on the presentations at the conference, San Antonio, TX, USA, March 7–10, 2010. New York, NY: Springer (ISBN 978-1-4614-0771-3/hbk; 978-1-4614-0772-0/ebook). Springer Proceedings in Mathematics 13, 345-365 (2012).
In this article, which is of expository character, the author describes a method for transferring results from two model cases of compact plane sets , namely and the unit circle, to more general compact plane sets. The basic point is that many interesting properties of compact plane sets are preserved when taking polynomial inverse images.
For a polynomial let denote the inverse image of . This leads to the following method.
(a) Start from a result for the model case .
(b) Apply an inverse polynomial mapping to go to a special result on the inverse image of the model set .
(c) Approximate more general sets by inverse images as in (b).
Among others the polynomial inverse image method has been successful in the following situations:
– Bernstein-type inequalities, the model case being the classical Bernstein inequality on ;
– Markov-type inequalities, the model case being the classical Markov inequality on ;
– asymptotics of Christoffel functions on compact subsets of the real line, with model case ;
– asymptotics of Christoffel functions on curves, with model case ;
– universality on general sets, the model case being on ;
– fine zero spacing of orthogonal polynomials, with model case ;
– Bernstein-type inequalities for a system of smooth Jordan curves, the model case being Bernstein’s inequality on .