Approximation of functions and data

Abstract

Approximating a function f consists of replacing it by another function $\tilde{f}$ of simpler form that may be used as its surrogate. This strategy is used frequently in numerical integration where, instead of computing $\int_a^b {f(x)dx}$, one carries out the exact computation of $\int_a^b \tilde{f}(x)dx$, $\tilde{f}$ being a function simple to integrate (e.g. a polynomial), as we will see in the next chapter. In other instances the function f may be available only partially through its values at some selected points. In these cases we aim at constructing a continuous function $\tilde{f}$ that could represent the empirical law which is behind the finite set of data. We provide some examples which illustrate this kind of approach.