A reminder of polynomial interpolation
Function approximation

There are two main ways of approximating a function f with finite information: either, as with the Taylor polynomial, by a finite number of components in a base of a finite vector subspace, which we define in advance on the basis of known or assumed properties of f, or by the values of f at a finite number of abscissas.

In the second category, once the finite information is known, it is necessary to (approximately) reconstruct f from its values at the abscissae. So let n + 1 distinct abscissas x₀,..., x_n of the interval [a, b] in which f is to be approximated, and f_j: = f (x_j) be the corresponding values of f. It is logically assumed that f ∊ C [a, b], the linear space of continuous functions on the interval [a, b], already to guarantee the existence of its value at any point. The most natural way to solve this problem is by interpolation: find a...