Piecewise polynomial approximation

Abstract

Introduction Up to now, the focus of our discussion has been the question of approximation of a given function f , defined on an interval [ a, b ], by a polynomial on that interval either through Lagrange interpolation or Hermite interpolation, or by seeking the polynomial of best approximation (in the ∞-norm or 2-norm). Each of these constructions was global in nature, in the sense that the approximation was defined by the same analytical expression on the whole interval [ a, b ]. An alternative and more flexible way of approximating a function f is to divide the interval [ a, b ] into a number of subintervals and to look for a piecewise approximation by polynomials of low degree. Such piecewise-polynomial approximations are called splines , and the endpoints of the subintervals are known as the knots . More specifically, a spline of degree n, n ≥ 1, is a function which is a polynomial of degree n or less in each subinterval and has a prescribed degree of smoothness. We shall expect the spline to be at least continuous, and usually also to have continuous derivatives of order up to k for some k , 0 ≤ k n . Clearly, if we require the derivative of order n to be continuous everywhere the spline is just a single polynomial, since if two polynomials have the same value and the same derivatives of every order up to n at a knot, then they must be the same polynomial.