Geodesic
The order of approximation by partial sums of lacunary series in Banach spaces
Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 575-582

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Conditions are determined which must be imposed on an $n_k$ lacunary sequence and a $(C,1)$-basis in Banach. space in order that the analog of the theorem concerning best approximation by partial sums of lacunary trigonometrical series should hold. 
@article{MZM_1970_8_5_a4,
     author = {V. F. Gaposhkin},
     title = {The order of approximation by partial sums of lacunary series in {Banach} spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {575--582},
     publisher = {mathdoc},
     volume = {8},
     number = {5},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a4/}
}
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V. F. Gaposhkin. The order of approximation by partial sums of lacunary series in Banach spaces. Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 575-582. http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a4/