Geodesic
Orthogonal bases for $L^p$ spaces
Matematičeskie zametki, Tome 10 (1971) no. 4, pp. 375-385.

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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. 
@article{MZM_1971_10_4_a1,
     author = {S. F. Gerasimov},
     title = {Orthogonal bases for $L^p$ spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {375--385},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_4_a1/}
}
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S. F. Gerasimov. Orthogonal bases for $L^p$ spaces. Matematičeskie zametki, Tome 10 (1971) no. 4, pp. 375-385. http://geodesic.mathdoc.fr/item/MZM_1971_10_4_a1/