2 - Polynomials and spline functions as approximating tools

Summary

In real situations the problem of approximating the function f(i) consists of replacing it following a given rule by a closed (in different senses) function φ(t) from an a priori fixed set N and estimating the error. The final result and the difficulties in obtaining it depend heavily on the choice of the approximating set N and the method of approximation, i.e. rules which determine how the function φ corresponds to f.

In choosing the approximating set, besides ensuring the necessary precision, one also needs to have functions φ which are simple and easy to study and calculate. Algebraic polynomials and (in the periodic case) trigonometric polynomials possess the simplest analytic structure. The main focus in approximation theory since it became an independent branch of analysis has been directed at the problem of approximation by polynomials. But, around the 1960s, spline functions started to play a more prominent role in approximation theory. They have definite advantages, in comparison with polynomials, for computer realizations and, moreover, it turns out that they are the best approximation tool in many important cases.

In this chapter we give the general properties of polynomials and polynomial splines that it will be necessary to know in the rest of the book. Let us note first that this introductory material is different in nature for polynomials and splines. The reason lies not only in their differing structures but also in the fundamental difference in the linear methods used for polynomial or spline approximation. Considering polynomials (algebraic or trigonometric), our main attention is concentrated, besides the classical Chebyshev theorem, on the linear methods based on the Fourier series and their analogs.