Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$.
My question is the following:
Are there other examples of function $\psi(x)$, such that $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$ or, at least, orthogonal and complete system in $L^2([0,1])$?
It is known, for instance, that square-wave functions of the form $\mbox{sign}(\sin(n \pi x))$ is complete system in $L^2([0,1])$, however, it is not orthogonal. Of course, one can orthogonalize this system, but the result will be "non-periodic" in the sense that the elements of the orthogonalized set cannot be constructed by replications of the initial function $\mbox{sign}(\sin(\pi x))$.
By the similar reasons Haar system, Legendre polynomials, etc. are not candidates, too.
It seems to me that the question is rather natural and it should be some results on this. However, I cannot find anything. Need your advice!
It seems that I have found the answer in this article. The main result (restricted to the interval $(0, 1)$) is that only $$ \{\sin(n \pi x)\}_{n \in \mathbb{N}} \quad \mbox{or} \quad \{1, \cos(n \pi x)\}_{n \in \mathbb{N}} $$ forms a periodic orthogonal basis in $L^2(0, 1)$.
I'll leave this answer here for better search on this question.