An extremal property of Hermite polynomials

Let Hn be the nth Hermite polynomial, i.e., the nth orthogonal on R polynomial with respect to the weight w(x) = exp(−x2). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f | |Hn| at the zeros of Hn+1, then for k = 1, . . . , n we have f (k) H (k) n , where · is the L2(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 L2(w;R) norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.  2003 Elsevier Inc. All rights reserved.