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Christoffel functions on curves and domains
Author:
Vilmos Totik
Journal:
Trans. Amer. Math. Soc. 362 (2010), 2053-2087
MSC (2000):
Primary 26C05, 31A99, 41A10
Posted:
November 18, 2009
MathSciNet review:
2574887
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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  times the  -th Christoffel function tends to the Radon-Nikodym derivative of the measure with respect to the equilibrium distribution of the support of the measure.
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- 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
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- B. Gustafsson, M. Putinar, E. B. Saff and N. Stylianopoulos, Bergman polynomials on an archipalego: estimates, zeros and shape reconstruction (manuscript, arXiv0811.1715v1).
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- K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constructive Approximation, 6(1990), 1-20. MR 1027506 (90k:26023)
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- A. N. Kolmogorov, Stationary sequences in Hilbert spaces, Bull. Moscow State Univ., 2(1941), 1-40 (in Russian).
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- 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)
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- D. S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Annals of Mathematics (to appear).
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 for  and  bounds for orthogonal polynomials, Ann. of Math., 111(1980), 145-154. MR 558399 (81c:42003)
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- 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)
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- 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)
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- P. Nevai, Géza Freud, Orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory, 48(1986), 1-167. MR 862231 (88b:42032)
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- R. Nevanlinna, Analytic Functions, Grundlehren der mathematischen Wissenschaften, 162, Springer-Verlag, Berlin, 1970. MR 0279280 (43:5003)
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- V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math., 187 (2001), 139-160. MR 1864632 (2002h:41017)
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- V. Totik, Universality and fine zero spacing on general sets, Arkiv för Math., doi:10.1007/s11512-008-0071-3 (to appear)
- 39.
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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
Email:
totik@math.usf.edu
DOI:
http://dx.doi.org/10.1090/S0002-9947-09-05059-4
PII:
S 0002-9947(09)05059-4
Received by editor(s):
April 7, 2008
Posted:
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.
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