Approximation of functions and data

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 ∫ _a bf(x)dx one carries out the exact computation of $$ \int_{a}^{b} {\tilde{f}\left( x \right)dx} $$$$ \tilde f $$ 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 a couple of examples which illustrate this kind of approach.