Acta Mathematica Hungarica

, Volume 83, Issue 1–2, pp 27–58 | Cite as

Description of Extremal Polynomials on Several Intervals and their Computation. I

  • K. Schiefermayr
  • F. Peherstorfer
Article

Abstract

Let Et = ∪j=1l [a2j-1, a2j], a1 < a2 < ... < a2l. First we give a complete characterization of that polynomial of degree n which has n + l extremal points on El. Such a polynomial is called T-polynomial because it shares many properties with the classical Chebyshev polynomial on [−1,1], e.g., it is minimal with respect to the maximum norm on El, its derivative is minimal with respect to the L1-norm on El, etc. It is known that T-polynomials do not exist on every El. Then it is demonstrated how to generate in a very simple illustrative geometric way from a T-polynomial on l intervals a T-polynomial on l or more intervals. For the case of two and three intervals a complete description of those intervals on which there exists a T-polynomial is provided. Finally, we show how to compute T-polynomials by Newton's method.

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References

  1. [1]
    N. I. Achieser, Über einige Funktionen, die in gegebenen Intervallen am wenigsten von Null abweichen, Bull. Phys. Math., 3 (1929), 1–69.Google Scholar
  2. [2]
    N. I. Achieser, Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen, Bull. Acad. Sci. URSS, 7 (1932), 1163–1202.Google Scholar
  3. [3]
    A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours and periodic motions of Toda Lattices, Math. USSR Sbornik, 53 (1986), 233–260.Google Scholar
  4. [4]
    W. E. Ferguson Jr., H. Flaschka and D. W. McLaughlin, Nonlinear normal modes for the Toda chain, J. Comput. Physics, 45 (1982), 157–209.Google Scholar
  5. [5]
    B. Fischer, Chebyshev polynomials for disjoint compact sets, Constr. Approx., 8 (1992), 309–329.Google Scholar
  6. [6]
    H. Hochstadt, On the theory of Hill's matrices and related inverse spectral problems, Lin. Alg. Appl., 11 (1975), 41–52.Google Scholar
  7. [7]
    H. Hochstadt, An inverse spectral theorem for a Hill's matrix, Lin. Alg. Appl., 57 (1984), 21–30.Google Scholar
  8. [8]
    A. Kroó, On uniqueness of best L 1-approximation on disjoint intervals, Math. Z., 191 (1986), 507–512.Google Scholar
  9. [9]
    G. G. Lorentz, Approximations of Functions, Holt, Rinehart and Wista (New York, 1966).Google Scholar
  10. [10]
    F. Peherstorfer, On Tehebycheff polynomials on disjoint intervals, in: Colloq. Math. Soc. Bolyai, 49. Haar Mem. Conf. (Budapest, 1985), pp. 737–751.Google Scholar
  11. [11]
    F. Peherstorfer, Orthogonal polynomials in L 1-approximation, J. Approx. Th., 52 (1988), 241–268.Google Scholar
  12. [12]
    F. Peherstorfer, On Gauss quadrature formulas with equal weights, Num. Math., 52 (1988), 317–327.Google Scholar
  13. [13]
    F. Peherstorfer, Orthogonal and Chebyshev polynomials on two intervals, Acta Math. Hungar., 55 (1990), 245–278.Google Scholar
  14. [14]
    F. Peherstorfer, On Bernstein-Szegö orthogonal polynomials on several intervals II: Orthogonal polynomials with periodic recurrence coefficients, J. Approx. Th., 64 (1991), 123–161.Google Scholar
  15. [15]
    F. Peherstorfer, On orthogonal and extremal polynomials on several intervals, J. Comp. Appl. Math., 48 (1993), 187–205.Google Scholar
  16. [16]
    F. Peherstorfer, Elliptic orthogonal and extremal polynomials, Proc. London Math. Soc., 70 (1995), 605–624.Google Scholar
  17. [17]
    A. Pinkus and Z. Ziegler, Interlacing properties of the zeros of the error function in best L p-approximations, J. Approx. Th., 27 (1979), 1–18.Google Scholar
  18. [18]
    T. J. Rivlin, The Chebyshev Polynomials, Wiley & Sons (New York, 1974).Google Scholar
  19. [19]
    M. L. Sodin and P. M. Yuditskii, Algebraic solution of E. I. Zolotarev and N. I. Akbiezer problems on polynomials that deviate least from zero, Teor. Funktsii Funktsional Anal. i Prilozhen, 56 (1991), 56–64.Google Scholar
  20. [20]
    M. L. Sodin and P. M. Yuditskii, Functions deviating least from zero on closed subsets of the real axis, St. Petersburg Math. J., 4 (1993), 201–249.Google Scholar
  21. [21]
    M. Toda, Theory of Nonlinear Lattices, Springer Series in Solid State Sciences 20 (Berlin, 1984).Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • K. Schiefermayr
    • 2
  • F. Peherstorfer
    • 1
  1. 1.Institut für Analysis und NumerikJ. Kepler Universität LinzLinzAustria
  2. 2.Institut für Algebra, Stochastik und Wissensbasierte Mathematische SystemeJ. Kepler Universität LinzLinzAustria

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