Definition of polynomial interpolation

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.