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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.

41A10Approximation by polynomials
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
Full Text: DOI
[1] Aptekarev, A. I.: Asymptotic properties of polynomials orthogonal on a system of contours and periodic motions of Toda lattices, Math. USSR sb. 53, 223-260 (1986) · Zbl 0608.42016 · doi:10.1070/SM1986v053n01ABEH002918
[2] Bogatyrev, A. B.: Effective computation of Chebyshev polynomials for several intervals, Math. USSR sb. 190, 1571-1605 (1999) · Zbl 0986.34042 · doi:10.1070/SM1999v190n11ABEH000438
[3] Contests in higher mathematics (Hungary, 1949 -- 1961). In memoriam Miklós Schweitzer, Editorial Board: G. Szász, L. Gehér, I. Kovács, L. Pintér, Manuscript revised by P. Erdös, A. Rényi, B. Sz.-Nagy, P. Turán, Akadémiai Kiadó, Budapest, 1968, p. 260.
[4] A. Dold, Lectures on Algebraic Topology, Grundlehren der mathematischen Wissenschaften, vol. 200, Springer, Berlin, Heidelberg, New York, 1972. · Zbl 0234.55001
[5] Geronimo, J. S.; Vanassche, W.: Orthogonal polynomials on several intervals via a polynomial mapping, Trans. amer. Math. soc. 308, 559-581 (1988) · Zbl 0652.42009 · doi:10.2307/2001092
[6] Peherstorfer, F.: On Bernstein -- Szegő orthogonal polynomials on several intervals, II, J. approx. Theory 64, 123-161 (1991) · Zbl 0721.42017
[7] Peherstorfer, F.: Orthogonal and extremal polynomials on several intervals, J. comput. Appl. math. 48, 187-205 (1993) · Zbl 0790.42012 · doi:10.1016/0377-0427(93)90322-3
[8] Peherstorfer, F.: Deformation of minimizing polynomials and approximation of several intervals by an inverse polynomial mapping, J. approx. Theory 111, 180-195 (2001) · Zbl 1025.42014 · doi:10.1006/jath.2001.3571
[9] Ransford, T.: Potential theory in the complex plane, (1995) · Zbl 0828.31001
[10] E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, vol. 316, Springer, Berlin, Heidelberg, 1997. · Zbl 0881.31001
[11] K. Schiefermayr, Lower bound for the minimum deviation of the Chebyshev polynomials on several intervals (manuscript). · Zbl 1217.41031
[12] H. Stahl, V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics, vol. 43, Cambridge University Press, New York, 1992. · Zbl 0791.33009
[13] G. Székely (Ed.), Contests in Higher Mathematics, Problem Books in Mathematics, Springer, New York, 1995.
[14] Totik, V.: Polynomial inverse images and polynomial inequalities, Acta math. (Scandinavian) 187, 139-160 (2001) · Zbl 0997.41005 · doi:10.1007/BF02392833
[15] Totik, V.: The inheritance problem and monotone systems, Austral. math. Soc. gazette 33, 122-130 (2006)
[16] Widom, H.: Extremal polynomials associated with a system of curves in the complex plane, Adv. math. 3, 127-232 (1969) · Zbl 0183.07503 · doi:10.1016/0001-8708(69)90005-X