Description of extremal polynomials on several intervals and their computation. I

Let E_l = ∪_j=1^l [a_2j-i , a_2j], a₁ < a₂ < ... < a_2l. First we give a complete characterization of that polynomial of degree n which has n + l extremal points on E_l. 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_l, its derivative is minimal with respect to the L₁-norm on E_l, etc. It is known that T-polynomials do not exist on every E_l. 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.