Coherent pairs of measures and Markov-Bernstein inequalities

TL;DR

This paper explores all seven known coherent pairs of measures related to linear functionals, deriving three-term recurrence relations for associated polynomials, and analyzing the asymptotic behavior of the Markov-Bernstein constants linked to their smallest zeros.

Contribution

It provides explicit recurrence relations for polynomials associated with all coherent pairs and investigates the asymptotic properties of the Markov-Bernstein constants.

Findings

Seven types of recurrence relations are established.

Explicit bounds for the smallest zeros are derived.

Asymptotic behavior of the constants is characterized.

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 $\mu_{1,n}$ 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 $M_n=\frac{1}{\sqrt{\mu_{1,n}}}$. The seven kinds of three term recurrence relations are given. In the case where $c_0 =e^{-x} dx+\delta(0)$ and $c_1 =e^{-x} dx$, explicit upper and lower bounds are given for $\mu_{1,n}$, and the asymptotic behavior of the corresponding Markov-Bernstein constant is stated. Except in a part of one case, $\lim_{n \to \infty} \mu_{1,n}=0$ is proved in all the cases.