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

Denote by E a set of the form j=1 m [a j ,b j ] and let 𝒯 n =infx n + 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 Kcap(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:
41A10Approximation by polynomials
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
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