Coherent pairs of measures and Markov-Bernstein inequalities

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.Comment: 32 page

Semiclassical asymptotic behavior of orthogonal polynomials

Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and study the corresponding second order difference equation. We suggest an Ansatz for its solutions playing the role of the semiclassical Green-Liouville Ansatz for solutions of the Schr\"odinger equation. The formulas obtained for $P_{n}(z)$ as $n\to\infty$ generalize the classical Bernstein-Szeg\"o asymptotic formulas

Remez-type inequalities and their applications

AbstractThe Remez inequality gives a sharp uniform bound on [−1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [−1, 1], where |;p|; is at most 1, is known. Remez-type inequalities give bounds for classes of functions on a line segment, on a curve or on a region of the complex plane, given that the modulus of the functions is bounded by 1 on some subset of prescribed measure. This paper offers a survey of the extensive recent research on Remez-type inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and Müntz polynomials. Remez-type inequalities play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remez-type inequalities by giving a number of applications