Geodesic
Orthogonal bases in $L^p$
Izvestiya. Mathematics , Tome 4 (1970) no. 5, pp. 1169-1181.

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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 $L^\infty=C$. 
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B. V. Ryazanov; A. N. Slepchenko. Orthogonal bases in $L^p$. Izvestiya. Mathematics , Tome 4 (1970) no. 5, pp. 1169-1181. http://geodesic.mathdoc.fr/item/IM2_1970_4_5_a9/

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