Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights
Let \(\{p_ n: n= 0,1,\dots\}\) be a system of orthogonal polynomials with respect to the Freud weight \(e^{-x^ m}\) on the real line where \(m\) is an even positive integer. Denote by \(x_{kn}\), \(k= 1,2,\dots,n\), the zeros of \(p_ n\). For a function \(f\) on the real line the Hermite- Fejér interpolation polynomial \(L_ n(\nu; f,x)\) of order \(\nu= 1,2,\dots\) based at the zeros \(x_{1n},x_{2n},\dots, x_{nn}\) is uniquely defined to be the algebraic polynomial of degree \(\nu n- 1\) by the conditions \[ L_ n(\nu; f,x_{kn})= f(x_{kn}),\qquad k= 1,2,\dots,n; \] \[ L^{(r)}_ n(\nu; f, x_{kn})= 0,\quad k=1,2,\dots, n,\quad r= 1,2,\dots, r-1. \] For the H-F interpolation of even order the authors find an estimate for pointwise convergence rate which leads to the validity of the following relation for every uniformly continuous function \(f\): \[ \lim_{n\to\infty} \max_{M\leq x\leq M} | L_ n(2d; f,x)- f(x)|=0,\;\forall M> 0,\;d= 1,2,\dots\;. \] In the case of odd order, a divergence result is established which leads to the existence of such continuous function \(f\) that \[ \limsup_{n\to\infty} \max_{a\leq x\leq b} | L_ n(2d- 1; f,x)|= \infty \] for any \(a< b\).
scientific article; zbMATH DE number 650896