Abstract :
Denote by ηi=cos(iπ/n), i = 0, ..., n the extreme points of the Chebyshev polynomial Tn(x) = cos(n arc cos x). Let πn be the set of real algebraic polynomials of degree not exceeding n, and let Bn be the unit ball in the space πn equipped with the discrete norm |p|n,∞ ≔ max0 ≤ i ≤ n|p(ηi)|. We prove that the unique solutions of the extremal problems maxp ∈ Bn ∫1−1 [p(k + 1)(x)]2(1 − x2)k − 1/2dx, k = 0, ..., n − 1, and maxp ∈ Bn ∫1− 1[p(k + 2)(x)]2(1 − x2)k − 1/2dx, k = 0, ..., n − 2, are p(x) = ±Tn(x), and we obtain the extremal values in an explicit form.
