Geodesic
Certain exact bounds for seminorms given on spaces of periodic functions
Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 789-798.

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Certain new bounds are established for the values of seminorms given on the spaces $C$ and $L^p(1\le p\infty)$ of periodic functions by means of the norm of the function itself and its finite differences, as well as of the moduli of continuity. These bounds are applied to concrete seminorms; in particular, to the best approximation, which yields a refinement of the direct theorems in approximation theory. The results obtained for spaces $C$ and $L^1$ are exact. 
@article{MZM_1977_21_6_a5,
     author = {V. V. Zhuk},
     title = {Certain exact bounds for seminorms given on spaces of periodic functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {789--798},
     publisher = {mathdoc},
     volume = {21},
     number = {6},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_6_a5/}
}
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V. V. Zhuk. Certain exact bounds for seminorms given on spaces of periodic functions. Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 789-798. http://geodesic.mathdoc.fr/item/MZM_1977_21_6_a5/