This isn't an abstract question, but rather a request for someone to find a particular error. I keep getting the wrong answer, even though I have a similar answer to the correct one. It's not some wildly incorrect answer, which is perhaps most shocking.
Given the following: $$ h(x)=x^3; \langle f,g\rangle = \int_a^b(f\cdot g)(x)dx\space\space\forall\space f,g \in C[a,b]$$
find the best parabolic aproximation of $h(x)$. This problem is relatively straightforward, as I use the Legendre polynomials to form an orthogonal basis $\{1,x,x^2-\frac{1}{3}\}$ for $P_2$ and just project $h$ onto each basis vector. This is the exact procedure provided by the book for this section, so an alternative method is not likely. The problem is my answer is different from the answer in the back, and I am not sure if I am simply making a calculation error or a procedural error. Graphing my answer and the book's answer proves that theirs is clearly correct and my answer is not. Anyone feel like showing work?
I figured it out. The chosen basis $\{1,x,x^2-\frac{1}{3}\}$ is not orthogonal for $P_2[a,b]$ necessarily. It is only orthogonal for $a=-b$ (a symmetric interval over the inner product space). Use Gram Schmidt to orthogonalize the standard basis of $P_2$ using the new inner product we have defined.