In $C([0,1],R)$ consider the subspace from the orthogonal set {$1,\cos(2\pi x),\cos(4\pi x),..,\cos(2N\pi x)$} what func is closest to $\sin(\pi x) $

Question

In $C([0,1],R)$ consider the subspace from the orthogonal set : $\{1,\cos(2\pi x),\cos(4\pi x),...,\cos(2N\pi x) \}$

Find the function in the subspace which is as close to $\sin(\pi x) $ as possible

how do i solve this?

Answer 1

I you meant close to $f(x) =\sin(\pi x)$ in the $L^2([0,1])$ sense, then extend $f$ to a $1$-periodic even function,

$f$ is equal to its Fourier series $$f = a_0+\sum_{n=1}^\infty a_n \cos(2\pi n .)+b_n \sin(2\pi n .), \qquad \text{ in } L^2([0,1])$$

since $f$ is even, the $b_n = 0$, and hence $$f = a_0+\sum_{n=1}^\infty a_n \cos(2\pi n .) $$