Skip to main content
SpringerLink
Log in

Characterization of algebraic curves by Chebyshev quadrature

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

Complex potential theory is used to show that Chebyshev-type quadrature works particularly well on algebraic Jordan curves Γ in ℝd, supplied with normalized arc length or a similar probability measure μ. Evaluating the integral ∫Γ fdμ by the arithmetic mean of the value off on any cycle ofN equally spaced nodes on Γ (relative to μ), the quadrature error will, be bounded byAe −bN supΓ|f| for allN and all polynomialsf(x) of degree ≤cN. It is plausible that small shifts of the nodes would give quadrature error zero for such polynomials. There are related results for algebraic Jordan arcs and certain algebraic surfaces. The situation is completely different for nonalgebraic curves and surfaces, where corresponding quadrature remainders are at least of order 1/N.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Baran and W. Pleśniak,Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves, Studia Math.125 (1997), 83–96, MR98e:41016.

    MATH  MathSciNet  Google Scholar 

  2. T. Becker and V. Weispfenning,Gröbner bases, Graduate Texts in Math., Vol. 141, Springer-Verlag, New York, 1993, MR95e:13018.

    MATH  Google Scholar 

  3. S. N. Bernstein,On quadrature formulas with positive coefficients (Russian), Izv. Akad. Nauk SSSR, Ser. Mat.1 (1937), 479–503;Collected works, Vol. 2, Izdat. Akad. Nauk SSSR, Moscow, 1954, pp. 205–227, MR16:433.

    Google Scholar 

  4. L. Bos, N. Levenberg, P. Milman and B. A. Taylor,Tangential Markov inequalities characterize algebraic submanifolds of ℝN, Indiana Univ. Math. J.44 (1995), 115–138, MR96i:41009.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. M. Chirka,Complex Analytic Sets (Russian), Nauka, Moscow, 1985, MR86i:32002. [English translation:Complex Analytic Sets, Kluwer, Dordrecht, 1989.]

    MATH  Google Scholar 

  6. G. Freud,Orthogonale Polynome (German), Birkhäuser Verlag, Basel, 1969, MR58-1902 (Zbl169-80). [English translation:Orthogonal Polynomials, Pergamon Press, Oxford, 1971.]

    MATH  Google Scholar 

  7. M. Klimek,Pluripotential Theory, London Math. Soc. Monogr., N.s., Vol. 6, Oxford Univ. Press, 1991, MR93h:32021.

  8. J. Korevaar,Polynomial approximation numbers, capacities and extended Green functions for ℂand ℂN, inApproximation Theory V (C.K. Chui et al., eds.), Academic Press, New York, 1986, pp. 97–127, MR89b:32022.

    Google Scholar 

  9. J. Korevaar,Green functions, capacities, polynomial approximation numbers and applications in real and complex analysis, Nieuw Arch. Wisk., (4)4 (1986), 133–153, MR88a:32022.

    MATH  MathSciNet  Google Scholar 

  10. J. Korevaar,Chebyshev-type quadratures: use of complex analysis and potential theory, inComplex Potential Theory (P. M. Gauthier and G. Sabidussi, eds.), Kluwer, Dordrecht, 1994, pp. 325–364, MR96g:41029.

    Google Scholar 

  11. J. Korevaar and J. L. H. Meyers,Massive coalescence of nodes in optimal Chebyshev-type quadrature on [−1,1], Indag. Math. N.S.4 (1993), 327–338, Mr94h:41064.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. B. J. Kuijlaars,Chebyshev-type quadrature and partial sums of the exponential series, Math. Comp.64 (1995), 251–263, MR95c:65043.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. B. J. Kuijlaars,Chebyshev-type quadrature for analytic weights on the circle and the interval, Indag. Math., N.S.6 (1995), 419–432, MR96j:41027.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. G. Nevai,Orthogonal polynomials, Mem. Amer. Math. Soc., Vol. 18, 1979, no. 213, MR 80k:42025.

  15. W. Rudin,A geometric criterion for algebraic varieties, J. Math. Mech.17 (1967/68), 671–683, MR36-2829.

    MathSciNet  Google Scholar 

  16. A. Sadullaev,Estimates of polynomials on analytic sets (Russian), Izv. Akad. Nauk SSSR, Ser. Mat.46 (1982), 524–534, 671, MR83k:32018. [English translation USSR Izv.20 (1983), 493–502.]

    MATH  MathSciNet  Google Scholar 

  17. J. Siciak,On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc.,105 (1962), 322–357, MR26-1495.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Siciak,Extremal plurisubharmonic functions in ℂn, Ann. Polon. Math.39 (1981), 175–211, MR83e:32018.

    MATH  MathSciNet  Google Scholar 

  19. J. Siciak,Extremal plurisubharmonic functions and capacities in ℂn, Sophia Kokyuroku in Math. no. 14, Dept. of Math., Sophia Univ., Tokyo, 1982.

    Google Scholar 

  20. V. P. Zaharjuta,Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernstein-Walsh theorem for analytic functions of several complex variables (Russian), Ann. Polon. Math.33 (1976), 137–148, MR56-3333.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, Netherlands

    J. Korevaar

  2. Department of Mathematics, University of Calgary, T2N 1N4, Calgary, Alberta, Canada

    L. Bos

Authors
  1. J. Korevaar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Bos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Korevaar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korevaar, J., Bos, L. Characterization of algebraic curves by Chebyshev quadrature. J. Anal. Math. 75, 233–246 (1998). https://doi.org/10.1007/BF02788701

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02788701

Keywords

Search

Navigation