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 (1p<) 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 L¹ are exact.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 789–798, June, 1977.