Shchehlov M. Generalization and application of Whitney, Dzyadik and other inequalities for algebraic polynomials

The thesis study is devoted to the approximation of various functions by polynomials and related functions. The main goal of the research is to prove classical estimates of approximation errors by polynomials that preserve certain properties of the function, that is, provide the so­called constrained approximation. The special attention is paid to the study of Shape Preserving Approximations (abbreviation SPA), in particular comonotone and copositive approximations ­ that is, those that preserve intervals of monotonicity and, respectively, sign of the given function. The modern theory of SPA of functions continuous on a segment by algebraic polynomials is almost as complete as the corresponding theory of approximation without constraints. A natural extension of this theory is SPA of periodic functions, as well as SPA of functions of a complex variable, which are much more complicated. Nevertheless, during last 20­25 years there has been a significant advance of this theory, but, surprisingly enough, the final results have been obtained for piecewise convex (2­ monotone) and q­monotone functions, but not piecewise monotone (1­monotone ). Namely the piecewise monotone periodic functions have been investigated in the 6th section of the thesis. As for SPA functions of a complex variable, this section of the theory of functions is still at the beginning of its development and some progress is presented in Chapter 4 of the thesis. Chapter 5, dealing with the so­called hybrid polynomials, is complementary to Chapter 6, but has, in our view, its separate interest. Finally, Whitney inequality is an important tool in many proofs of function theory ­ in particular, in spline theory, constructive function theory, etc. A pointwise version of Whitney's inequality is studied in Chapter 3.