Acta Mathematica Hungarica

, Volume 158, Issue 1, pp 216–234 | Cite as

Christoffel functions on planar domains with piecewise smooth boundary

  • A. PrymakEmail author
  • O. Usoltseva


Up to a constant factor, we compute the Christoffel function on planar domains with boundary consisting of finitely many C2 curves such that each corner point of the boundary has interior angle strictly between 0 and \(\pi \). The resulting formula uses the distances from the point of interest to the curves or certain parts of the curves defining the boundary of the domain.

Key words and phrases

Christoffel function algebraic polynomial orthogonal polynomial boundary effect 

Mathematics Subject Classification

42C05 41A17 41A63 26D05 42B99 


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We are grateful to the referee for the careful reading of the manuscript and several valuable suggestions that pointed out some inaccuracies and, more importantly, led to an improvement of the generality of the result.


  1. 1.
    P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag (New York, 1995)Google Scholar
  2. 2.
    Cohen, A., Davenport, M.A., Leviatan, D.: On the stability and accuracy of least squares approximations. Found. Comput. Math. 13, 819–834 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, A., Migliorati, G.: Optimal weighted least-squares methods. SIAM J. Comput. Math. 3, 181–203 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ditzian, Z., Prymak, A.: On Nikol'skii inequalities for domains in \(\mathbb{R}^d\). Constr. Approx. 44, 23–51 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jackson, D.: On the application of Markoff's theorem to problems of approximation in the complex domain. Bull. Amer. Math. Soc. 37, 883–890 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kroó, A.: Christoffel functions on convex and starlike domains in \(\mathbb{R}^d\). J. Math. Anal. Appl. 421, 718–729 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kroó, A.: Multivariate "needle" polynomials with application to norming sets and cubature formulas. Acta Math. Hungar. 147, 46–72 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kroó, A.: Multivariate fast decreasing polynomials. Acta Math. Hungar. 149, 101–119 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math. (2), 170 (2009), 915–939Google Scholar
  10. 10.
    P. Nevai, Géza Freud, orthogonal polynomials and Christoffel functions. A  case study, J. Approx. Theory, 48 (1986), 3–167Google Scholar
  11. 11.
    L. A. Pastur, Spectral and probabilistic aspects of matrix models, in: Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Math. Phys. Stud. vol. 19, Kluwer (Dordrecht, 1996)Google Scholar
  12. 12.
    Prymak, A.: Upper estimates of Christoffel function on convex domains. J. Math. Anal. Appl. 455, 1984–2000 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Prymak and O. Usoltseva, Pointwise behavior of Christoffel function on planar convex domains, in: Topics in Classical and Modern Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser (accepted)Google Scholar
  14. 14.
    B. Simon, The Christoffel–Darboux kernel, in: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc. (Providence, RI, 2008), pp. 295–335Google Scholar
  15. 15.
    Videnskii, V.S.: Extremal estimates for the derivative of a trigonometric polynomial on an interval shorter than its period. Soviet Math. Dokl. 1, 5–8 (1960)MathSciNetGoogle Scholar
  16. 16.
    Walther, G.: On a generalization of Blaschke's rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22, 301–316 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Xu, Y.: Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal. 3, 257–272 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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