A discrete weighted Markov-Bernstein inequality for sequences and polynomials

Author(s)

Dimitrov, Dimitar K. ^[1] ; Nikolov, Geno P. ^[2]

Affiliation

^[1] Univ Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

^[2] Sofia Univ St Kliment Ohridski, Fac Math & Informat, 5 James Bourchier Blvd, Sofia 1164 - Bulgaria

• Document type: Journal article
• Source: Journal of Mathematical Analysis and Applications; v. 493, n. 1 JAN 1 2021.
• Web of Science Citations: 0

Abstract

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. (AU)

• FAPESP's process: 16/10357-1 - Zeros of orthogonal polynomials and special functions and inequalities of Markov type
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• FAPESP's process: 16/09906-0 - Harmonic analysis, approximation theory and applications
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Projects - Thematic Grants