Derivation of the errors involved in interpolation and their application to numerical quadrature formulae (Q1298610)

Let \(L_n(x)\) be the \(n\) th-order interpolation polynomial for a function \(f(x)\) defined on \([a, b]\) and based on the nodal points \(x_0,x_1,\cdots ,x_n\) with \(x_0=a<x_1<x_2<\cdots <x_n=b.\) A new approach to derive a closed-form expression for the error term \(E_n(x)=f(x)-L_n(x)\) is given. The upper estimates of the quantity \(|E_n(x)|\) for \(p (p\leq n)\) times differentiable functions are established. The problem is studied also for functions which are not differentiable at all. As applications of the received estimates, the corresponding errors for the quadrature formulae are derived. In addition a few examples for numerical experiments are considered.