Constants in Jackson-type inequations for the best approximation of periodic differentiable functions

Let us consider the space of continuous $2\,\pi$-periodic functions endowed with the uniform norm. The structural properties of the functions are commonly characterized by moduli of continuity of various orders. In 1911, D. Jackson established a number of fundamental theorems that give estimates for the best approximation by the modulus of continuity of the first order for the function and its derivatives. These results were later extended to the case when the estimates of the best approximations are produced by the moduli of continuity of arbitrary order. Inequalities of this type play an important role in the theory of approximation and are studied (in various ways) in a large number of works of many authors. Similar relations are called direct theorems of approximation theory or generalized Jackson inequalities. In this paper for a wide class of spaces new estimates were obtained for the constants in the generalized Jackson inequality for differentiable functions, in some cases, improving the previously known. The basic apparatus used in the work is approximation methods, which are constructed on the basis of V. A. Steklov functions. Bibliogr. 12.