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Chebyshev constants and the inheritance problem. (English) Zbl 1190.41002

Denote by $E$ a set of the form ${\bigcup }_{j=1}^{m}\left[{a}_{j},{b}_{j}\right]$ and let ${𝒯}_{n}=inf{\parallel {x}^{n}+\cdots \parallel }_{E}$ be the $n$-th Chebyshev constant for $E$. One of the main purposes of the paper is to give a new proof of the estimate ${𝒯}_{n}\le K\text{cap}{\left(E\right)}^{n}$, where $K$ does not depend on $n$. In addition, another result is given concerning the approximation of $E$ by polynomial inverse images of $\left[-1,1\right]$ with order $1/n$. The two theorems above are interrelated and arise from the new approach introduced by the author.

This approach is based on the statement in the so-called inheritance problem, and has the advantage of avoiding the appearance of $c$-intervals, with the technical difficulties that they entail. A section is devoted to give another application of the latter problem in a more general setting.

MSC:
 41A10 Approximation by polynomials 31A15 Potentials and capacity, harmonic measure, extremal length (two-dimensional)