Distribution of points for convergent interpolatory integration rules on (−∞, ∞)

Abstract

Letw be a “nice” positive weight function on (−∞, ∞), such asw(x)=exp(−⋎x⋎^α) α>1. Suppose that, forn≥1,

$$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$

is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1,

$$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$

Moreover, suppose that the sequence of rules {I _n} ^t8 _n=1 isconvergent:

$$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$

for all continuousf:R→R satisfying suitable integrability conditions.

What then can we say about thedistribution of the points {x _jn} ⁿ _j=1 ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI _n has precision other thann-1.