An extremal property of Hermite polynomials

Let H-n be the nth Hermite polynomial, i.e., the nth orthogonal on R polynomial with respect to the weight w(x) = exp(-x(2)). We prove the following: If f is an arbitrary polynomial of degree at most n, such that \f\ less than or equal to \H-n\ at the zeros of Hn+1, then for k = 1,..., n we have parallel tof((k))parallel to less than or equal to parallel toH(n)((k))parallel to, where parallel to (.) parallel to is the L-2(w; R) norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the L-2(w; R) norm, and estimates for the expansion coefficients in the basis of Hermite polynomials. (C) 2003 Elsevier Inc. All rights reserved.