期刊
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
卷 493, 期 1, 页码 -出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2020.124522
关键词
Markov-Bernstein inequality; Discrete inequality; Meixner weight; Meixner polynomials; Orthogonal polynomial; Chebyshev polynomial
资金
- Brazilian foundations CNPq [306136/2017-1]
- FAPESP [2016/09906-0, 2016/10357-1]
- Bulgarian National Research Fund [DN 02/14]
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.
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.
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