Extremal polynomials for weighted Markov inequalities (Q1100655)

Let \(\Pi^*_ n\) denote the class of all real monic polynomials of degree \(\leq n\) and let w be a positive weight function defined on (- \(\infty,\infty)\) with continuous derivative there. For each \(n=1,2,...\), define \(\lambda_ n=\sup \{\| wp'\| /\| wp\|:p\in \Pi^*_ n\}\) and \(\mu_ n=\sup \{\| (wp)'\| /\| wp\|:p\in \Pi^*_ n\}\) where the prime denotes differentiation and \(\| \cdot \|\) denotes the sup-norm on (-\(\infty,\infty)\). In this setting, alternation information for \(wp^*\) is developed where \(p^*\) is an extremal polynomial for \(\lambda_ n\) or \(\mu_ n\). In addition, for the special cases that \(w(x)=\exp (-x^ 2)\) and \(p^*\in \Pi^*_ n\) is an extremal for \(\mu_ n\), it is shown that \(wp^*\) is extremal.