Abstract
Given a non-polar compact set K,we define the n-th Widom factor W n (K) as the ratio of the sup-norm of the n-th Chebyshev polynomial on K to the n-th degree of its logarithmic capacity. By G. Szegő, the sequence \((W_{n}(K))_{n=1}^{\infty }\) has subexponential growth. Our aim is to consider compact sets with maximal growth of the Widom factors. We show that for each sequence \((M_{n})_{n=1}^{\infty }\) of subexponential growth there is a Cantor-type set whose Widom’s factors exceed M n . We also present a set K with highly irregular behavior of the Widom factors.
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Achieser, N.I.: Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen I. Bull. Acad. Sci. URSS 7(9), 1163–1202 (1932). (in German)
Achieser, N.I.: Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. II. Bull. Acad. Sci. URSS VII. Ser., 309–344 (1933). (in German)
Faber , G.: Über Tschebyscheffsche Polynome. J. für die Reine und Angewandte Math. 150, 79–106 (1920). (in German)
Fekete, M.: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17, 228–249 (1923). (in German)
Goncharov, A.P.: Weakly Equilibrium Cantor-type Sets. Potential Anal. 40, 143–161 (2014)
Peherstorfer, F.: Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math. 48, 187–205 (1993)
Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)
Schiefermayr, K.: A Lower Bound for the Minimum Deviation of the Chebyshev Polynomials on a Compact Real Set. East J. Approximations 14, 223–233 (2008)
Szegő, G.: Bemerkungen zu einer Arbeit von Herrn M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 21, 203–208 (1924). (in German)
Totik, V.: Chebyshev constants and the inheritance problem. J. Approximation Theory 160, 187–201 (2009)
Totik, V.: The norm of minimal polynomials on several intervals. J. Approximation Theory 163, 738–746 (2011)
Totik, V.: Chebyshev Polynomials on Compact Sets. Potential Anal. 40, 511–524 (2014). doi:10.1007/s11118-013-9357-6.
Widom, H.: Extremal Polynomials Associated with a System of Curves in the Complex Plane. Adv. Math. 3, 127–232 (1969)
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Goncharov, A., Hatinoğlu, B. Widom Factors. Potential Anal 42, 671–680 (2015). https://doi.org/10.1007/s11118-014-9452-3
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DOI: https://doi.org/10.1007/s11118-014-9452-3