Hermite and Hermite-Fejér interpolation and associated product integration rules on the real line: The L∞ theory

Abstract

We investigate convergence in a weighted L ∞-norm of Hermite-Fejér and Hermite interpolation and related approximation processes, when the interpolation points are zeros of orthogonal polynomials associated with weights W 2 = e −2 Q on the real line. For example, if H n(W 2,ƒ,x) denotes the nth Hermite-Fejér interpolation polynomial for W 2 = e −2 Q and the function ƒ, then we show that lim n→∞ xϵR { sup|H n(W 2, f, x)−f(x)| W 2(x)[1+|Q′(x)|] −k(1+|x|) −1}=0 under suitable conditions on ƒ, W 2, and κ. The weights to which the results are applicable include W 2(x) = exp(−¦x¦ α) , α > 1, or W 2(x) = exp(− exp k(¦x¦ α)) , α > 1, k ⩾ 1, where exp k denotes the kth iterated exponential. Convergence of product integration rules induced by the various approximation processes is then deduced. Essentially the conclusion of the paper is that by damping the error in approximation of ƒ Hermite-Fejér or Hermite interpolation by a factor [1 + ¦Q′(x)¦] −κ(1 + ¦x¦) −1 , which decays much more slowly than the weight W 2, we can ensure sup-norm convergence under quite general conditions.