A discrete weighted Markov-Bernstein inequality for sequences and polynomials

For parameters c is an element of(0,1) and beta > 0, let l(2)(c ,beta) be the Hilbert space of real functions defined on N (i.e., real sequences), for which parallel to f parallel to(2)(c,beta) := Sigma(infinity)(k=0)(beta)(k)/k! c(k)[f(k)](2) < infinity. We study the best (i.e., the smallest possible) constant gamma(n)(c,beta) in the discrete Markov-Bernstein inequality parallel to Delta P parallel to(c,beta) <= gamma(n)(c ,beta) parallel to P parallel to(c,beta), P is an element of P-n, where P-n is the set of real algebraic polynomials of degree at most n and Delta f(x) := f(x+1)-f(x). We prove that (i) gamma(n)(c, 1) <= 1 + 1/root c for every n is an element of N and lim(n ->infinity) gamma(n)(c, 1) = 1+1/root c; (ii) For every fixed c is an element of(0,1), gamma(n)(c, beta) is a monotonically decreasing function of beta in (0,infinity); (iii) For every fixed c is an element of(0,1) and beta > 0, the best Markov-Bernstein constants gamma(n)(c,beta) are bounded uniformly with respect to n. A similar Markov-Bernstein inequality is proved for sequences, and a relation between the best Markov-Bernstein constants gamma(n)(c, beta) and the smallest eigenvalues of certain explicitly given Jacobi matrices is established. (c) 2020 Elsevier Inc. All rights reserved.

Automated summary

This study investigates the optimal constants of discrete Markov-Bernstein inequalities for specified parameters, and explores their properties. Three conclusions about these constants were proven, demonstrating their behavior under different conditions. Additionally, a similar inequality was proved for sequences, establishing a relationship between the optimal constants and the smallest eigenvalues of Jacobi matrices.