Bezkryla S. On high orders moduli of continuity and p-monotonic approximation

The thesis is devoted to the study of some questions on the moduli of continuity of higher and fractional orders generated by a semigroup of operators, and also to construction of counterexamples for piecewise p-monotonic, (р>4 or р=4) approximation by algebraic polynomials. We give the definition of moduli of continuity of fractional order, of the semigroups of operators, their examples and elementary properties, and an example of an inequality for moduli of continuity of order a that holds for integers a and does not hold for non-integers a. The main result of the second section of the thesis is the proof that not every k-majorant is a modulus of continuity of the k -th order for (k>3 or k=3), but also the transfer of this assertion to moduli of continuity of fractional order. This result is a generalization of the corresponding statement for k=2, obtained by S. V. Konyagin for uniform moduli of continuity. The key moment in its proof is the establishment of some auxiliary inequality of a special kind. In the thesis, an auxiliary inequality is obtained for the modulus of continuity of order a in the case when there is not necessarily an integer a. The strengthening of this inequality is also established for the k-th modulus of continuity in the case natural k, (k>2 or k=2). In turn, for the third and fourth uniform moduli of continuity, we obtain a refinement of the general inequality for the k-th modulus of continuity. The proofs of these refined inequalities use the "classical" definition of the uniform modulus of continuity, which is given by finite differences of the third and fourth orders. A new counterexample is constructed showing that for a piece-wise p-monotonic, (р>4 or р=4) approximation by algebraic polynomials an inequality of Jackson-Stechkin type with the derivative r>p or r=p does not hold even with a constant depending on the approximated function. This result strengthens the well-known result of L. P. Yushchenko and generalizes for the case (р>4 or р=4) the corresponding result of the paper by D. Leviathan and I. A. Shevchuk obtained for p=3.