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)



Christoffel functions on curves and domains

Author: Vilmos Totik
Journal: Trans. Amer. Math. Soc. 362 (2010), 2053-2087
MSC (2000): Primary 26C05, 31A99, 41A10
Published electronically: November 18, 2009
MathSciNet review: 2574887
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Asymptotics for Christoffel functions are established for measures supported on unions of smooth Jordan curves and for area-like measures on unions of smooth Jordan domains. For example, in the former case $ n$ times the $ n$-th Christoffel function tends to the Radon-Nikodym derivative of the measure with respect to the equilibrium distribution of the support of the measure.

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

  • 1. T. Carleman, Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat. Astr. Fys., 17(1923), 215-244.
  • 2. L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies, 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986 (37:1576)
  • 3. W. Blaschke, Kreis und Kugel, Walter de Gruyter & Co., Berlin, 1956. MR 0077958 (17:1123d)
  • 4. M. Findley, Universality for regular measures satisfying Szegő's condition locally, J. Approx. Theory, 155, 136-154.
  • 5. G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1971.
  • 6. L. Golinskii, The Christoffel function for orthogonal polynomials on a circular arc, J. Approx. Theory, 101(1999), 165-174. MR 1726450 (2001b:42032)
  • 7. G. Golub, B. Gustafsson, P. Milanfar, M. Putinar and J. Varah, Shape reconstruction from moments: theory, algorithms and applications, SPIE Proceedings, Vol. 4116(2000), Advanced Signal Processing, Algorithms, Architecture and Implementations X (Franklin T. Luk, ed.), 406-416.
  • 8. U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, University of California Press, Berkeley and Los Angeles, 1958. MR 0094840 (20:1349)
  • 9. B. Gustafsson, C. He, P. Milanfar and M. Putinar, Reconstructing planar domains from their moments, Inverse Problems, 16(2000), 1053-1070. MR 1776483 (2001k:44010)
  • 10. B. Gustafsson, M. Putinar, E. B. Saff and N. Stylianopoulos, Les polynômes orthogonaux de Bergman sur un archipel, C. R. Acad. Sci. Paris, Ser. I XXX (2008). MR 2412785
  • 11. B. Gustafsson, M. Putinar, E. B. Saff and N. Stylianopoulos, Bergman polynomials on an archipalego: estimates, zeros and shape reconstruction (manuscript, arXiv0811.1715v1).
  • 12. K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constructive Approximation, 6(1990), 1-20. MR 1027506 (90k:26023)
  • 13. A. N. Kolmogorov, Stationary sequences in Hilbert spaces, Bull. Moscow State Univ., 2(1941), 1-40 (in Russian).
  • 14. M. G. Krein, Generalization of investigations by G. Szegő, V. I. Smirnov and A. N. Kolmogorov, Dokl. Akad. Nauk. SSSR, 46(1945), 91-94 (in Russian). MR 0013457 (7:156b)
  • 15. A. L. Levin, E. B. Saff and N. Stylianopoulos, Zero distribution of Bergman orthogonal polynomials for certain planar domians, Constr. Approx., 19(2003), 411-435. MR 1979059 (2004a:30007)
  • 16. D. S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Annals of Mathematics (to appear).
  • 17. 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., 111(1980), 145-154. MR 558399 (81c:42003)
  • 18. A. Máté, P. Nevai and V. Totik, Szegő's extremum problem on the unit circle, Annals of Math., 134(1991), 433-453. MR 1127481 (92i:42014)
  • 19. E. Miña-Díaz, E. B. Saff and N. S. Stylianopoulos, Zero distributions for polynomials orthogonal with weights over certain planar regions, Comput. Methods Funct. Theory, 5(2005), 185-221. MR 2174353 (2006f:30004)
  • 20. B. Nagy and V. Totik, Sharpening of Hilbert's lemniscate theorem, J. D´Analyse Math., 96(2005), 191-223. MR 2177185 (2006g:30008)
  • 21. P. Nevai, Géza Freud, Orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory, 48(1986), 1-167. MR 862231 (88b:42032)
  • 22. R. Nevanlinna, Analytic Functions, Grundlehren der mathematischen Wissenschaften, 162, Springer-Verlag, Berlin, 1970. MR 0279280 (43:5003)
  • 23. L. A. Pastur, Spectral and probabilistic aspects of matrix models. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 207-242, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. MR 1385683 (97b:82060)
  • 24. Ch. Pommerenke, On the derivative of a polynomial, Michigan Math. J., 6(1959), 373-375. MR 0109208 (22:95)
  • 25. Ch. Pommerenke, Boundary Behavior of Conformal Mappings, Grundlehren der mathematischen Wissenschaften, 299, Springer-Verlag, Berlin, Heidelberg, New York, 1992. MR 1217706 (95b:30008)
  • 26. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995 MR 1334766 (96e:31001)
  • 27. F. Riesz and M. Riesz, Über die Randwerte einer analytischen Funktion, Quatrieme Congrés de Math. Scand., 1916.
  • 28. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, 316, Springer-Verlag, New York/Berlin, 1997. MR 1485778 (99h:31001)
  • 29. B. Simon, Weak convergence of CD kernels and applications, Duke Math. J., 146(2009), no. 2, 305-330. MR 2477763
  • 30. B. Simon, The Christoffel-Darboux kernel, ``Perspectives in PDE, Harmonic Analysis and Applications'' in honor of V.G. Maz'ya's 70th birthday, to be published in Proceedings of Symposia in Pure Mathematics
  • 31. B. Simon, Two extensions of Lubinsky's universality theorem, J. D´Analyse Math., 105(2008), 345-362 MR 2438429
  • 32. B. Simon, Orthogonal Polynomials on the Unit Circle, V.1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005. MR 2105088 (2006a:42002a)
  • 33. H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications, 43, Cambridge University Press, Cambridge, 1992. MR 1163828 (93d:42029)
  • 34. G. Szegő, Orthogonal Polynomials, Coll. Publ., XXIII, Amer. Math. Soc., Providence, 1975.
  • 35. G. Szegő, Collected Papers, ed. R. Askey, Birkhaüser, Boston-Basel-Stuttgart, 1982.
  • 36. V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. D'Analyse Math., 81 (2000), 283-303. MR 1785285 (2001j:42021)
  • 37. V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math., 187 (2001), 139-160. MR 1864632 (2002h:41017)
  • 38. V. Totik, Universality and fine zero spacing on general sets, Arkiv för Math., doi:10.1007/s11512-008-0071-3 (to appear)
  • 39. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, third edition, Amer. Math. Soc. Colloquium Publications, XX, Amer. Math. Soc., Providence, 1960. MR 0218587 (36:1672a)
  • 40. A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1959. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26C05, 31A99, 41A10

Retrieve articles in all journals with MSC (2000): 26C05, 31A99, 41A10

Additional Information

Vilmos Totik
Affiliation: Bolyai Institute, Analysis Research Group of the Hungarian Academy os Sciences, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary – and – Department of Mathematics, University of South Florida, 4202 E. Fowler Avenue, PHY 114, Tampa, Florida 33620-5700

Received by editor(s): April 7, 2008
Published electronically: November 18, 2009
Additional Notes: The author was supported by NSF DMS 0700471
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society