Find the minimum coefficients in an inner product on L2(-1,1) using Legendre polynomials as orthonormal vectors.

Question

Given that the orthonormal vectors in $L^{2}$(-1,1) obtained by applying the Gram-Schmidt process to 1,x,$x^{2}$ are scalar multiples of the first three Legendre polynomials: $P_{0}(x) = 1$, $P_{1}(x) = x$, $P_{2}(x) = 1/2(3x^{2}-1)$. Find: $$\min_{a,b,c \in \mathbb C} \int^{1}_{-1}|x^{3}-a-bx-cx^{2}|^{2}dx$$

So I think a start would be to get it into the form of the inner product with respect to the orthonormal vectors but I'm not too sure how to go about it, any hints would be much appreciated.

Answer 1

Your task is to project $p(x)=x^3$ onto the subspace of quadratic polynomials, for which the polynomials $P_0,P_1,P_2$ provide an orthonormal basis. The projection is $$q = \sum_{k=1}^3 \langle p,P_k\rangle P_k$$ and I leave it for you to compute $\int |p-q|^2$.