Acta Mathematica Hungarica

, Volume 83, Issue 1, pp 27–58

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

  • K. Schiefermayr
  • F. Peherstorfer

DOI: 10.1023/A:1006607401740

Cite this article as:
Schiefermayr, K. & Peherstorfer, F. Acta Mathematica Hungarica (1999) 83: 27. doi:10.1023/A:1006607401740


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.

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