Let El = ∪j=1l [a2j-i , 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.
|Number of pages||32|
|Journal||Acta Mathematica Hungarica|
|Publication status||Published - Apr 1999|