A reminder of polynomial interpolation

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 x ₀ ,..., 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...