The Bergman Shift Operator on Polynomial Lemniscates

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• 作者：Brian Simanek (1)
  
• 关键词：Weak asymptotic measures ; Ratio asymptotics ; Bergman Polynomials ; Bergman shift operator ; Hessenberg matrices ; Christoffel transform ; Primary 42C05 ; Secondary 60B10 ; 15B05 ; 47B35
• 刊名：Constructive Approximation
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：41
• 期：1
• 页码：113-131
• 全文大小：236 KB
• 参考文献：1. Baratchart, L., Saff, E.B., Stylianopoulos, N.: On finite-term recurrence relations for Bergman and Szeg艖 polynomials. Comput. Methods Funct. Theory 12(2), 393鈥?02 (2012) CrossRef
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  6. Putinar, M., Stylianopoulos, N.: Finite-term relations for planar orthogonal polynomials. Complex Anal. Oper. Theory 1(3), 447鈥?56 (2007) CrossRef
  7. Saff, E.B., Stahl, H., Stylianopoulos, N., Totik, V.: Orthogonal polynomials for area-type measures and image recovery. arXiv:1043.6456
  8. Saff, E.B., Stylianopoulos, N.: Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions. Complex Anal. Oper Theory 8(1), 1鈥?4 (2014) CrossRef
  9. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields, Grundlehren der Mathematischen Wissenschaften, Band 316, Springer, Berlin (1997)
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  12. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part One: Classical Theory. American Mathematical Society, Providence (2005)
  13. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part Two: Spectral Theory. American Mathematical Society, Providence (2005)
  14. Simon, B.: The Christoffel-Darboux kernel. In: Perspectives in PDE, Harmonic Analysis, and Applications, A Volume in Honor of V. G. Maz鈥檡a鈥檚 70th Birthday, Proceedings Symposium Pure Mathematics, vol .79, pp. 295鈥?35 (2008)
  15. Simon, B.: Szeg艖鈥檚 Theorem and its Descendants: Spectral Theory for \(L^2\) perturbations of Orthogonal Polynomials. Princeton University Press, Princeton (2010)
  16. Stylianopoulos, N.: Bergman Polynomials on Archipelaga: Recent Developments on Theory and Applications, plenary address given at CMFT 2013 Conference. www2.ucy.ac.cy/nikos/CMFT2013.pdf
  17. Widom, H.: Extremal polynomials associated with a system of curves and arcs in the complex plane. Adv. Math. 3, 127鈥?32 (1969) CrossRef
  
• 作者单位：Brian Simanek (1)
  
  1. Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

We investigate the relationship between the Bergman shift operator and the support of the corresponding measure. We pay special attention to the situation when the measure of orthogonality is concentrated on a polynomial lemniscate. As an application of our new results, we obtain a ratio asymptotic result for a wide variety of measures supported on polynomial lemniscates.