On some modifications of Jackson's generalized theorem for the best approximations of periodic functions

Let us consider the space of continuous 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 is studied (variously ways) in a large number of works of many authors. Similar relations are called direct theorems or generalized Jackson inequalities. Inequalities that contain the estimates of norm for intermediate derivative by the rules norms of the function and its derivatives of a higher order than the ones being estimated also play an important role in approximation theory. They are called the inequalities of Landau–Kolmogorov. In this paper the non-standard modification of Jackson-type inequalities with respect to the direction suggested by inequalities of Landau–Kolmogorov are obtained for a wide class of spaces obtained. The main instruments used in the work are approximation methods built on the basis of functions of V. A. Steklov. Bibliogr. 8.