Is the family $\{ \sin(kx ) \}$ a complete system in $L_{2}(0,1)$ for every $ k \in \mathbb{R}$, $k\geq1$ ? And less specifically, Is it a complete system in $L_{2}(0,b)$ for every b$\in \mathbb{R}$ ?
I am trying to obtain a solution for a PDE using the Fourier method (Separation of Variables) but I need to obtain $b_{k}$ for :
$ \sum b_{k} \sin(k\pi x)=f(x) $
And I am interested in using the formula:
$b_k = \frac {\int_{0}^{1} f(x) \varphi_k (x) dx}{\int_{0}^{1} \varphi_{k} (x)^{2} dx }$