Acta Mathematica Hungarica

, Volume 83, Issue 1–2, pp 27–58

# 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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