2017/11/20 by Geno Nikolov, Nikolov, Geno, Rumen Uluchev +1
Mathematics · #41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1711.07398
openalex publication_date 2017/11/20 · openalex created_date 2017/12/04 · openalex updated_date 2026/07/28
We prove two-sided estimates for the best (i.e., the smallest possible) constant cn(α) in the Markov inequality ‖pn'‖wα ≤ cn(α) ‖pn‖wα , pn ∈ \cal Pn . Here, \cal Pn stands for the set of algebraic polynomials of degree ≤ n, wα(x) := xα e-x, α> -1, is the Laguerre weight function, and ‖⋅‖wα is the associated L2-norm, ‖f‖wα = (∫0∞ |f(x)|2 wα(x) dx)1/2 . Our approach is based on the fact that cn-2(α) equals the smallest zero of a polynomial Qn, orthogonal with respect to a measure supported on the positive axis and defined by an explicit three-term recurrence relation. We employ computer algebra to evaluate the seven lowest degree coefficients of Qn and to obtain thereby bounds for cn(α). This work is a continuation of a recent paper [5], where estimates for cn(α) were proven on the basis of the four lowest degree coefficients of Qn.