Polynomial interpolation: definition

Given $$n +1$$ points $$(x_k,f_k)$$ with $$k\in\{0,1,\dots,n\}\ $$ and $$\ x_k\neq x_i \ $$ if $$\ i\neq k$$, we call polynomial interpolation when determining a polynomial of lower degree or same degree $$n$$ such that $$p(x_k)=f_k \ $$ for every $$\ k$$.

This polynomial always exists and it is unique.

Sometimes we calculate the interpolating polynomial of a set of data. In other occasions, the values $$f_k$$ correspond to the value of a certain function $$f(x)$$ in the points $$x_k$$. Namely, instead of working with the very function, sometimes it is more comfortable to work with a polynomial similar enough. But how similar to the original function will this polynomial be? This is quantified by the interpolation error:

$$$ \text{error}=|f(x)-P_n(x)|=$$$ $$$=\Big| \dfrac{f^{(n+1)}(\xi(x))}{(n+1)!}(x-x_0)\cdot(x-x_1) \cdots (x-x_n)\Big|$$$

where $$\xi$$ is a point belonging to the interval generated by all the points $$x_k$$.

It is necessary to say that the function must be at least $$n +1$$ times derivable.

As we already said, the polynomial is unique, but there are several methods to calculate it.