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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 \(C_1\).
For the entire collection see [Zbl 1230.65002].

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
26D05 Inequalities for trigonometric functions and polynomials
30C10 Polynomials and rational functions of one complex variable
30C85 Capacity and harmonic measure in the complex plane
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References:

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