The basisness of $\sin(nx)$ in $L^2(0,\pi)$ is a very classical fact.
I wonder whether the system of absolute values $|\sin(nx)|$ is also a basis (or, at least, a complete system) in $L^2(0,\pi)$. Clearly, every $|\sin(nx)|$ is an even function with respect to $x=\pi/2$. In particular, such functions cannot approximate a noneven function, and, consequently, the restriction to half-interval $(0,\pi/2)$ is necessary.
Thus, my precise question can be formulated as follows.
Do the functions $|\sin(nx)|$ form a basis (or, at least, a complete system) in $L^2(0,\pi/2)$?
Perhaps, the answer to this simply-stated question is well known, but I was not able to find it in the literature. Any hint will be appreciated.