Geodesic
A theorem on the approximation of a trigonometric sum by a shorter one
Izvestiya. Mathematics , Tome 71 (2007) no. 2, pp. 341-370

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We prove a theorem on the approximation of a trigonometric sum by a shorter one with the constants in the remainder calculated concretely. 
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     title = {A theorem on the approximation of a trigonometric sum by a shorter one},
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A. A. Karatsuba; M. A. Korolev. A theorem on the approximation of a trigonometric sum by a shorter one. Izvestiya. Mathematics , Tome 71 (2007) no. 2, pp. 341-370. http://geodesic.mathdoc.fr/item/IM2_2007_71_2_a3/