Oscillating Polynomials of Least L₁-Norm

Abstract

A classical way to construct a quadrature formula is to approximate the integral $\int\limits_a^b {f\left( t \right)}$dt by $\int\limits_a^b {H\left( {,f;t} \right)dt}$ where H(x, f; t) is the Hermite interpolation polynomial for the function f based on a given system of nodes x = {(x₁, v₁),..., (x_n, v_n)} (i.e. x₁,...,x_n of multiplicities v₁,...,v_n, respectively). The error E(x; f) of this quadrature rule is expressed by the divided difference f[x,x] of f at the points x and x. We have

$$E\left( {;f} \right) = \int\limits_a^b {f\left[ {,x} \right]{{\left( {x - {x_1}} \right)}^{{v_1}}}...{{\left( {x - {x_n}} \right)}^{{v_n}}}dx} $$