# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${E}_{0}=\left[-1,1\right]$ 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 ${T}^{-1}\left({E}_{0}\right)$ 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 $E={T}^{-1}\left({E}_{0}\right)$ 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 $\left[-1,1\right]$;

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

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

– asymptotics of Christoffel functions on curves, with model case ${C}_{1}$;

– universality on general sets, the model case being on $\left[-1,1\right]$;

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

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

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