Consider the inner product space $ C[0,2\pi] $ (or its completion, the Hilbert Space $ L^2[0,2\pi] $) as in 3.5.1 Find the values of $ c_1, c_2, $ and $ c_3 $ which minimize the value of $$ \int_0^{2\pi} |t-(c_1\sin(t)+c_2\sin(2t)+3\sin(3t))|^2 dt \text{.} $$
Any help/hints would be greatly appreciated.
The sin functions are an orthonormal basis $\{s_n\}$of the subspace of odd functions, so $t = \sum_{n=1}^{\infty} a_n s_n $. We want to minimise $|\sum_{n=4}^{\infty} a_n s_n +(a_1-c_1)s_1+(a_2-c_2)s_2+(a_3-c_3)s_3|^2$. Using orthonormality leads to $a_i = c_i$ .