CHAPTER 3 - CONVERGENCE OF POLYNOMIAL INTERPOLATION

Abstract

This chapter discusses convergence of polynomial interpolation. In the approximation of a function f (x) by a linear combination of a given set of functions ϕi (x), the uniform or Chebyshev norm is the most powerful of all the Holder norms. In trigonometric polynomials, each ϕi is either of the form cos rx or of the form sin rx. The required result is then one of the standard results of the theory of Fourier series. The corresponding result for polynomials is Weierstrass' theorem. The chapter presents a proof for Weierstrass' theorem. A polynomial approximation Bn (f) exists of any prescribed accuracy. The construction of Bn (f) is very simple, and suggests that it might be a useful practical approximation and a means of proving Weierstrass' theorem. A more useful practical method of obtaining a polynomial approximation to f(x) is to use the Lagrangian polynomial, which takes the same value as f(x) at a number of selected points x1, x2 , … , xn in the range [a, b].