Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights

Abstract

We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials onℝ thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e ^−Q, whereQ:ℝ→ℝ is even and continuous inℝ, Q^" is continuous in (0, ∞) andQ ^'>0 in (0, ∞), while, for someA, B,

$$1< A \leqslant \frac{{(d/dx)(xQ'(x))}}{{Q'(x)}} \leqslant B,x \in (0,\infty )$$

Leta _n denote thenth Mhaskar-Rahmanov-Saff number forQ, andL>0. Then, uniformly forn≥1 and |x|≤a _n (1+Ln ^−2/3),

$$\lambda _n (W^2 ,x) \sim \frac{{a_n }}{n}W^2 (x)\left( {\max \left\{ {n^{ - 2/3} ,1 - \frac{{|x|}}{{a_n }}} \right\}} \right)^{ - 1/2}$$

Moreover, for all x εℝ, we can replace ∼ by ≥. In particular, these results apply toW(x):=exp(-|x|^α), α>1. We also obtain lower bounds for allx εℝ, when onlyA>0, but this necessarily requires a more complicated formulation.

We deduce that thenth orthonormal plynomialp _n (W ², ·). forW ² satisfies

$$\mathop {\sup }\limits_{x \in \mathbb{R}} |p_n (W^2 ,x)|W(x)\left| {1 - \frac{{|x|}}{{a_n }}} \right|^{1/4} \sim a_n^{ - 1/2}$$

and

$$\mathop {\sup }\limits_{x \in \mathbb{R}} |p_n (W^2 ,x)|W(x) \sim a_n^{ - 1/2} n^{1/6} .$$

In particular, this applies toW(x):=exp(-|x|^α), α>1.