Orthogonal projections over the space of linear functions.

Question

I've encountered the following exercise:

In the space $C[0,2\pi]$ with internal product $\langle f,g\rangle = \int_0^{2\pi} fg$, determine the trigonometric polynomial of $a+b\cos(x) + c\sin(x)$ which better aproximates the function $f(x) = x$.

I know this could be solved by calculating the orthonormal projection of the trigonometric polynomial over $f(x)$, but I don't know how that'd be calculated, as I can't see any basis (let alone any orthonormal basis for $f(x) = x$).

Answer 1

HINT

You are suggested to pick the approximating function which is in the span of the basis $\mathcal{B} = \{1, \sin x, \cos x\}$ already.

In a different view of your question, you are asked to provide the projection of $f(x)=x$ onto the vector space $span(\mathcal{B})$.

Answer 2

Hint:

You need to find an orthonormal basis $\{f_1, f_2, f_3\}$ for the subspace $\operatorname{span}\{1, \sin x, \cos x\}$ and then your desired function will be $$\sum_{i=1}^3 \langle f, f_i\rangle f_i$$

Check that we can take $$\{f_1, f_2, f_3\} = \left\{\frac1{\sqrt{2\pi}}, \frac{\cos x}{\sqrt{\pi}}, \frac{\sin x}{\sqrt{\pi}}\right\}.$$