Hermite Interpolation Of Higher Order And Convergence Of Gaussian Quadrature Formulas On Infinite Intervals

In 1878 in his fundamental work [12] Ch. Hermite introduced the so called Hermite interpolation. A lot of papers deal with this topic, but only few of them investigate Hermite interpolation of higher order. Most of these papers appeared in the last ten years and many important developments were obtained. There are two lines of study for the Hermite interpolation: the special system of nodes and an arbitrary system of nodes.In this paper, more stronger estimations of bounds for the fundamental functions of Hermite interpolation of higher order on an arbitrary system of nodes on infinite intervals are given. Based on this result, convergence of Gaussian quadrature formulas for Riemann-Stieltjes integrable functions on an arbitrary system of nodes on infinite intervals is discussed.In the first chapter, we list some known results of Hermite interpolation of higher order and Gaussian quadrature formulas as its simple survey and introduction.In the second chapter, we study a basic theorem of estimations for the fundamental functions of Hermite interpolation of higher order on infinite intervals.In the third chapter, first we state some useful lemmas, then we study the convergence of Gaussian quadrature formulas.In the last chapter we list some open problems.