Orthogonal bases in $L^p$

The following theorem is proved: Given any interval $I\subset[1,2)$, there is an orthonormal system $\{\varphi_n\}$ defined on $[0,1]$ which is a basis in $L^p$ for all $p\in I$, but is not a basis in $L^q$ for any $q\in[1,\infty]\setminus I$. Here $L^\infty=C$.