Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fine asymptotics of Christoffel functions for general measures

Author: Elliot Findley
Journal: Trans. Amer. Math. Soc. 363 (2011), 4553-4568
MSC (2010): Primary 30E10
Published electronically: April 19, 2011
MathSciNet review: 2806683
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mu$ be a measure on the unit circle satisfying Szegő's condition. In 1991, A. Máté calculated precisely the first-order asymptotic behavior of the sequence of Christoffel functions associated with $ \mu$ when the point of evaluation lies on the circle, resolving a long-standing open problem. We extend his results to measures supported on smooth curves in the plane. In the process, we derive new asymptotic estimates for the Cesáro means of the corresponding 1-Faber polynomials and investigate some applications to orthogonal polynomials, linear ill-posed problems and the mean ergodic theorem.

References [Enhancements On Off] (What's this?)

  • 1. A. Bakushinsky and A. Goncharsky. Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers, Boston, 1994. MR 1325921 (96d:65101)
  • 2. L. Golinskii. The Christoffel Function for Orthogonal Polynomials on a Circular Arc. J. Approx. Theory 101, pp. 165-174 (1999). MR 1726450 (2001b:42032)
  • 3. U. Grenander and G. Szegő. Toeplitz Forms and their Applications. University of California Press, Los Angeles, 1958. MR 0094840 (20:1349)
  • 4. N. Korneichuk. Exact Constants in Approximation Theory. Cambridge University Press, New York, 1991. MR 1124406 (92m:41002)
  • 5. A. Máté and P. Nevai. Bernstein's inequality in $ L^p$ for $ 0<p<1$ and $ (C,1)$ bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), 145-154. MR 558399 (81c:42003)
  • 6. A. Máté, P. Nevai and V. Totik. Szegő's Extremum Problem on the Unit Cricle, Ann. of Math. (2) 134 (1991), 433-453. MR 1127481 (92i:42014)
  • 7. Ch. Pommerenke. Boundary Behavior of Conformal Mappings, Grundlehren der mathematischen Wissenschaften, 299, Springer-Verlag, Berlin, Heidelberg, New York, 1992. MR 1217706 (95b:30008)
  • 8. T. Ransford. Potential Theory in the Complex Plane. Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • 9. W. Ross and W. Wogen. Common Cyclic Vectors for Normal Operators, Ind. Univ. Math. J. 53 (2004), 1537-1550. MR 2106335 (2005h:47041)
  • 10. W. Rudin. Real and Complex Analysis, 3rd ed., Mc-Graw Hill, Singapore, 1987. MR 924157 (88k:00002)
  • 11. P. K. Suetin. Fundamental Properties of Polynomials Orthogonal on a Contour. Russian Math. Surveys 21 (1968), 35-83. MR 0198111 (33:6270)
  • 12. V. Totik. Asymptotics for Christoffel functions for general measures on the real line. J. Anal. Math. 81 (2000), 283-303. MR 1785285 (2001j:42021)
  • 13. V. Totik. Christoffel functions on curves and domains. Trans. Amer. Math. Soc. 362 (2010), 2053-2087. MR 2574887 (2011b:30006)
  • 14. A. Zygmund. Trigonometric Series, Vols. I and II, 2nd ed., Cambridge Univ. Press, Cambridge, 1979. MR 0236587 (38:4882)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30E10

Retrieve articles in all journals with MSC (2010): 30E10

Additional Information

Elliot Findley

Received by editor(s): October 7, 2008
Received by editor(s) in revised form: March 2, 2009, April 20, 2009, and May 21, 2009
Published electronically: April 19, 2011
Additional Notes: This research was partially supported by NSF grant NSF DMS 0700471.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society