Abstract
Abstract Let Wα,ρ = xα(1 – x2)ρe–Q(x), where α > – $\begin{array}{} \displaystyle \frac12 \end{array}$ and Q is continuous and increasing on [0, 1), with limit ∞ at 1. This paper deals with orthogonal polynomials for the weights $\begin{array}{} \displaystyle W^2_{\alpha, \rho} \end{array}$ and gives bounds on orthogonal polynomials, zeros, Christoffel functions and Markov inequalities. In addition, estimates of fundamental polynomials of Lagrange interpolation at the zeros of the orthogonal polynomial and restricted range inequalities are obtained.
Highlights
Introduction and resultsIn this paper, for α > −, we setWα,ρ(x) = xα( − x )ρW(x), x ∈ [, ), for which the moment problem possesses a unique solution, and discuss the orthogonal polynomials for the weight Wα,ρ on [, )
This paper deals with orthogonal polynomials for the weights Wα,ρ and gives bounds on orthogonal polynomials, zeros, Christo el functions and Markov inequalities
The main results tell us that adding an even factor ( − x ) ρ to the weight x α e− Q(x), α > −, under su cient conditions for ρ and Q(x), its properties will be invariant
Summary
Such W is called an exponential weight on I. Levin and Lubinsky [3, 4] discussed orthogonal polynomials for exponential weights W on [− , ] and (c, d), c < < d, respectively. Liu and Shi [6] considered generalized Jacobi-Exponential weights UW, where U(x) is generalized Jacobi weights on (c, d), c < < d, and gave the estimates of the zeros of orthogonal polynomials for UW. The following theorems are similar in spirit with their analogues for weights ( − x )ρ e−Q(x) on two-sided intervals [12]. (b) There exists n such that uniformly for n > n , ≤ j ≤ n, pn Wα,ρ (xjn) ∼ φn(xjn)− xjn(an − xjn) − ,.
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