Orthogonal bases for $L^p$ spaces

The spectrum of a system of functions which are orthogonal on $[0, 1]$ is the set of all $p\in[1,\infty]$ such that the system forms a basis in $L^p[0, 1]$ ($L^\infty=C$). A set $E$ is called a \underbar{spectral set} if there exists a system of functions orthonormal on $[0, 1]$ whose spectrum is $E$. In this note we determine all spectral sets and construct an orthonormal system corresponding to each of them.