Franz Peherstorfer, Klaus Schiefermayr,
"Description of extremal polynomials on several intervals and their computation I"
, in Acta Mathematica Hungarica, Vol. 83, Seite(n) 27-58, 1999
Description of extremal polynomials on several intervals and their computation I
Sprache des Titels:
Let E be the union of l intervals. First we give a complete characterization of that polynomial of degree n, which has n+l extremal points on E. 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 E, its derivative is minimal with respect to the L1-norm on E, etc. It is known that not on every E there exists a T-polynomial. 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.