Certain exact bounds for seminorms given on spaces of periodic functions

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.