Geodesic
A periodic in the mean extension and bases of exponential functions in $L^p(-\pi,\pi)$
Matematičeskie zametki, Tome 12 (1972) no. 1, pp. 37-42.

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We develop sufficiency conditions for: 1) periodic-in-the-mean extendability of functions from $L^p$; 2) a system of exponential functions to be a basis in $L^p(-\pi,\pi)$. 
@article{MZM_1972_12_1_a4,
     author = {A. M. Sedletskii},
     title = {A periodic in the mean extension and bases of exponential functions in $L^p(-\pi,\pi)$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {37--42},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_1_a4/}
}
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A. M. Sedletskii. A periodic in the mean extension and bases of exponential functions in $L^p(-\pi,\pi)$. Matematičeskie zametki, Tome 12 (1972) no. 1, pp. 37-42. http://geodesic.mathdoc.fr/item/MZM_1972_12_1_a4/