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Chebyshev constants and the inheritance problem. (English) Zbl 1190.41002
Denote by $E$ a set of the form $\bigcup_{j=1}^m [a_j,b_j]$ and let $\mathcal{T}_n=\inf\|x^n+\dots\|_E$ 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 $\mathcal{T}_n \leq K \text{cap}(E)^n$, where $K$ does not depend on $n$. In addition, another result is given concerning the approximation of $E$ by polynomial inverse images of $[-1,1]$ 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)
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##### References:
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