# Markov-type inequalities and extreme zeros of orthogonal polynomials

ASCW03 - Approximation, sampling, and compression in high dimensional problems

The talk is centered around the problem of finding (obtaining  tight two-sided bounds for)  the sharp constants in certain Markov-Bernstein type inequalities in weighted $L_2$ norms. It turns out that, under certain assumptions, this problem is equivalent to the estimation of the extreme zeros of orthogonal polynomials with respect to a measure supported on $R_{+}$. It will be shown how classical tools like the Euler-Rayleigh method and Gershgorin circle theorem produce surprisingly good bounds for the extreme zeros of the Jacobi, Gegenbauer and Laguerre polynomials. The sharp constants in the $L_2$  Markov inequalities with the Laguerre and Gegenbauer weight functions and in a discrete $\ell_2$ Markov-Bernstein inequality are investigated using the same tool.

This talk is part of the Isaac Newton Institute Seminar Series series.