Polynomial Interpolation

The exact values of a unknown function $f$ are given:

$$\left[\begin{array}{cccc}{x}_0& x_1& \dots & x_n\\ y_0& y_1& \dots & y_n\end{array}\right]$$

$x_i\in \mathbb{R},y_i\in \mathbb{R}$

interpolant: a function $\tilde{f}$ such that $\tilde{f}(x_i)=y_i$ (the function we need to find)

Suppose $f\in V$ of a polynomial vector space $\tilde{f}\in V_n$ where $V_n$ is a subspace of $V$ with $\mathbf{dim}{V}_n=n$

$$f(x)\approx f_n(x)=\sum _{j=1}^n{a}_j{b}_j(x)$$