A convergence theorem for the Pál method of interpolation on the roots of Hermite polynomials
Let \(H_ n\) be the Hermite polynomial of degree n and let \(x_{i,n}\) \((i=1,2,...,n)\), \(x^*_{i,n}\) \((i=1,2,...,n-1)\) be the roots of the polynomials \(H_ n\) and \(H_ n'\), respectively. If \(f\in C^ 1(R)\) is such that \(f(0)=0\), \(\lim_{| x| \to +\infty}x^{2r}e^{-x^ 2/2} f(x)=0\) \((r=1,2,...)\) and \(\lim_{| x| \to +\infty}f'(x)e^{-x^ 2/2}=0,\) then for the sequence \((R_ n)\), \(n=2,4,6,..\). of interpolation polynomials of degree n, verifying \(R_ n(x_{i,n})=f(x_{i,n})\), \(R_ n'(x^*_{i,n})=f'(x^*_{i,n})\), the relation \(e^{-\nu /w^ 2}| f(x)-R_ n(x)| =O(\omega (f',n^{-1/2})\log n)\) is satisfied, where \(\nu >1\) and O does not depend on x.
scientific article; zbMATH DE number 3977589
scientific article