Widom Factors

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• 作者：Alexander Goncharov ; Burak Hatino?lu
• 关键词：Logarithmic capacity ; Chebyshev numbers ; Cantor sets ; 31A15 ; 30C85 ; 41A50 ; 28A80
• 刊名：Potential Analysis
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：42
• 期：3
• 页码：671-680
• 全文大小：196 KB
• 参考文献：1.Achieser, N.I.: über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen I. Bull. Acad. Sci. URSS 7(9), 1163-202 (1932). (in German)
  2.Achieser, N.I.: über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. II. Bull. Acad. Sci. URSS VII. Ser., 309-44 (1933). (in German)
  3.Faber , G.: über Tschebyscheffsche Polynome. J. für die Reine und Angewandte Math. 150, 79-06 (1920). (in German)
  4.Fekete, M.: über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17, 228-49 (1923). (in German)View Article MATH MathSciNet
  5.Goncharov, A.P.: Weakly Equilibrium Cantor-type Sets. Potential Anal. 40, 143-61 (2014)View Article MATH MathSciNet
  6.Peherstorfer, F.: Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math. 48, 187-05 (1993)View Article MATH MathSciNet
  7.Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)View Article MATH
  8.Schiefermayr, K.: A Lower Bound for the Minimum Deviation of the Chebyshev Polynomials on a Compact Real Set. East J. Approximations 14, 223-33 (2008)MATH MathSciNet
  9.Szeg?, G.: Bemerkungen zu einer Arbeit von Herrn M. Fekete: über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 21, 203-08 (1924). (in German)View Article MathSciNet
  10.Totik, V.: Chebyshev constants and the inheritance problem. J. Approximation Theory 160, 187-01 (2009)View Article MATH MathSciNet
  11.Totik, V.: The norm of minimal polynomials on several intervals. J. Approximation Theory 163, 738-46 (2011)View Article MATH MathSciNet
  12.Totik, V.: Chebyshev Polynomials on Compact Sets. Potential Anal. 40, 511-24 (2014). doi:10.-007/?s11118-013-9357-6.-/span> View Article MATH MathSciNet
  13.Widom, H.: Extremal Polynomials Associated with a System of Curves in the Complex Plane. Adv. Math. 3, 127-32 (1969)View Article MATH MathSciNet
  
• 作者单位：Alexander Goncharov (1)
  Burak Hatino?lu (2)
  
  1. Department of Mathematics, Bilkent University, 06800, Ankara, Turkey
  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Potential Theory
  Probability Theory and Stochastic Processes
  Geometry
  Functional Analysis
  
• 出版者：Springer Netherlands
• ISSN：1572-929X

文摘

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. Szeg?, 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.