Consider $2$-norm approximation of the function $f (x) = e^x$ for$ x ∈ [−1, 1]$ by a polynomial $QN(x)$ of degree $N$ which is expanded into a series of Legendre polynomials $Pj(x):$
$QN (x) = \sum (λjPj (x))$ Calculate the expansion coefficients $λ_0, λ_1, λ_2$ and $λ_3$.
This question was previously asked by another person on stack exchange, I've copied their answer as a new post because I wasn't sure how to 'bump' a post. I don't understand how I'd calculate the coefficients and also don't understand what it means by a 2-norm approximation of the function.
Edit: