Lagrange basis function is for the set of all polynomials of a given degree?

Question

Wikipedia says that

The set $\{(t−1)(t−2)/2, −t(t−2), t(t−1)/2\}$ is another basis for quadratic polynomials, called the Lagrange basis.

I do not understand the concept of Lagrange basis functions, because they appear only when talking about a set of points and doing interpolation. So, I can define a Lagrange basis function only when I have a set of points. Now, wikipedia does not provide this set of points (although it can be noticed that it is $\{0,1,2\}$) and simply says that are basis functions.

1. Does it mean that I can write any polynomial of degree maximum $2$ with these three Lagrangian basis functions?

2. Those three functions don't have nothing to do with the points $\{0,1,2\}$?

3. If I were to consider other three points (let's say $\{1,2,3\}$) and compute the Lagrangian basis functions, can I write any polynomial of degree maximum $2$ with these three new Lagrangian basis functions?

Answer 1

Let $S_n=\{0,\dots, n\}$.

$\forall i\neq j \in S_n$, let $L_i$ be the Lagrange polynomial so that $L_i(i) = 1$ and $L_i(j)=0$. As a reminder:

$$L_i(x) = \prod_{j\in S_n, j\neq i} \frac{x-x_j}{x_i-x_j}$$

1) Then, the Lagrange interpolation theorem indeed assesses that $(L_i)_{i\in S_n}$ is a basis of $\mathbb{R}_n[X]$ and that the coefficients of the decomposition of any polynomial $P$ in that basis is $(P(i))_{i\in S_n}$.

2) Those three functions are the $L_i(x)$ (ie. the Lagrange polynomial corresponding to the point $i$ of the set $S_n$).

3) The Lagrange interpolation theorem assesses that for any set $S \subset \mathbb{R}$ so that $|S|=n+1$, the Lagrange polynomials built from $S$ are a basis of $\mathbb{R}_n[X]$ and that the decomposition of $P$ is $(P(x))_{x\in S}$.