I have some linear function $f(x) = ax + b$ and I am trying to find a polynomial $p \in \mathbb{P^3}$ such that $p(0),p'(0) = 0$ and $\int_0^1 |f(x) - p(x)|^2dx$ is minimized. This means $p = ax^3 + bx^2$ with $a,b\in\mathbb{(F=R)}$
I've tried to start with this, but then am completely lost. Let $\mathbb{C_r}[0,1]$ be real, non-negative functions on $[0,1]$ with inner product $<f(x),g(x)> = \int_0^1 f(x)g(x)dx$.
Thanks :)
Let $M=\{cx^{2}+dx^{3}: c,d \in \mathbb R \}$. Them $M$ is a closed subspace of $L^{2}[0,1]$ and the functions $g(x)=x^{2},h(x)=x^{3}-\frac 5 6 x^{2}$ are orthogonal in the space. Choose $\alpha, \beta$ such that $\alpha g$ and $\beta h$ have norm $1$. Then all that you have to do is to project the given function $ax+b$ on to $M$ and the answer is $\langle (ax+b), \alpha g \rangle \alpha g+\langle (ax+b), \beta h \rangle \beta h$.