*(English)*Zbl 1268.41008

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 ${T}^{-1}\left({E}_{0}\right)$ 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 $[-1,1]$;

– 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:

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 |