Extremal Properties of Ultraspherical Polynomials

AbstractFor the polynomials Pl(x) = al0 + al1x + ... + allxl (of degree l) we consider the problem of maximizing a weighted product of the absolute values of the highest coefficients ∏nl = 1 |allβl among all polynomials P1, ..., Pn for which the weighted sum of squares ∑nl = 1 βlP2l(x) is bounded by 1 on the interval [−1, 1]. By an application of a duality result the solutions (depending on the weights βl ≥ 0) of these problems are determined. The "optimal" polynomials are the orthonormal polynomials with respect to a probability measure minimizing a weighted product of determinants of Hankel matrices (the solution of the dual problem). For a special class of weights β1, ..., βn the optimal polynomials can be represented in terms of ultraspherical polynomials. Thus some new extremal properties are obtained for these polynomials which generalize the well known fact that among all polynomials Pn of degree n with |Pn(x)| ≤ 1 (on [−1, 1]) the maximum of the highest coefficient is obtained for the Chebyshev polynomial of the first kind. The results are illustrated in several examples