Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I

For Jacobi parameters belonging to one of three classes: asymptotically periodic, periodically modulated, and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel–Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports.

Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)

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.