Petrova I. "Interpolatory estimates for comonotone approximation

In the first section, the approximation by monotone piecewise polynomial continuous functions (splines) of monotonic functions on a segment is considered. First, we prove a negative result, that for each natural r and for each partition of the segment there is a monotonic infinitely differentiable function on this segment such that any continuous spline of degree r cannot interpolate at the end of any part of the line. if it interpolates the function itself at the ends of the segment (see Theorem 1.2.1). This negative result leads to the assumption that approximation of monotone functions by monotone splines is not possible under the condition of high order of interpolation. However, the next is positive the result refutes this assumption (see Theorem 1.2.4). It turns out that each monotonic function of high smoothness can be approximated arbitrarily by these splines with a high order of interpolation if the diameters of the partitions are smaller than a sufficiently small constant H, which depends on the function. The main result of Section 1 is Theorem 1.1.1. In Subsection 1.4 it is shown that in the case of a monotonic approximation the analogues of the results of [12] obtained for the case of a convex approximation are also valid in the corresponding more accurate form (see Theorems 1.4.4 and 1.4 .5). In the second section, we consider the approximation of smooth convex functions f to an interval by convex algebraic polynomials that interpolate f and its derivatives at the endpoints of this interval. The main result of Section 2 is Theorem 2.12. One of the important consequences of the basic The theorem is a statement for any convex on [-1; 1] of the function f from the Sobolev space W ^ r. In the third section it is shown that (1) is incorrect, generally speaking, c the number N is independent of f i for fractional derivatives of order r> 3 (see Theorem 3.3.1 and Theorem 3.4.1). The main novelty of the study is to obtain interpolation estimates for monotonic and convex approximation of functions. The dissertation has a theoretical character, but as well as other results of this direction can have practical applications.