Coherent pairs of measures and Markov-Bernstein inequalities

Abstract

All the coherent pairs of measures associated to linear functionals $c_0$ and $c_1$, introduced by Iserles et al in 1991, have been given by Meijer in 1997. There exist seven kinds of coherent pairs. All these cases are explored in order to give three term recurrence relations satisfied by polynomials. The smallest zero $μ_{1,n}$ of each of them of degree $n$ has a link with the Markov-Bernstein constant $M_n$ appearing in the following Markov-Bernstein inequalities: $$ c_1((p^\prime)^2) \le M_n^2 c_0(p^2), \quad \forall p \in \mathcal{P}_n, $$ where $M_n=\frac{1}{\sqrt{μ_{1,n}}}$. The seven kinds of three term recurrence relations are given. In the case where $c_0 =e^{-x} dx+δ(0)$ and $c_1 =e^{-x} dx$, explicit upper and lower bounds are given for $μ_{1,n}$, and the asymptotic behavior of the corresponding Markov-Bernstein constant is stated. Except in a part of one case, $\lim_{n \to \infty} μ_{1,n}=0$ is proved in all the cases.