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The polynomial inverse image method. (English) Zbl 1268.41008
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 E 0 , namely E 0 =[-1,1] and E 0 =C 1 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 T let T -1 (E 0 ) denote the inverse image of E 0 . This leads to the following method.

(a) Start from a result for the model case E 0 .

(b) Apply an inverse polynomial mapping to go to a special result on the inverse image E=T -1 (E 0 ) of the model set E 0 .

(c) Approximate more general sets by inverse images E 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 [-1,1];

– Markov-type inequalities, the model case being the classical Markov inequality on [-1,1];

– asymptotics of Christoffel functions on compact subsets of the real line, with model case [-1,1];

– asymptotics of Christoffel functions on curves, with model case C 1 ;

– universality on general sets, the model case being on [-1,1];

– fine zero spacing of orthogonal polynomials, with model case [-1,1];

– Bernstein-type inequalities for a system of smooth Jordan curves, the model case being Bernstein’s inequality on C 1 .

MSC:
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
26D05Inequalities for trigonometric functions and polynomials
30C10Polynomials (one complex variable)
30C85Capacity and harmonic measure in the complex plane