An extremal property of Hermite polynomials. (Q1428231)

Consider the sequence of the Hermite polynomials \(H_m(x)=(-1)^m e^{x^2}\frac{d^m}{dx^m} \{e^{-x^2}\}\), \((m=0,1,\dots)\), i.e. orthogonal polynomials in \(L_2(\mathbb R,w)\), where \(w(x)=\exp(-x^2)\). The main result of the paper is the following Duffin-Schaeffer type inequality. If \(f\) is a polynomial on \(\mathbb R\) of degree at most \(n\) such that \(| f| \leq| H_n| \) at the zeros of \(H_{n+1}\), then for \(k=1,\dots,n\), \(\| f^{(k)}\| \leq\| H_n^{(k)}\| .\) Moreover, equality in the above inequality holds if and only if \(f=cH_n\) with \(| c| =1\).