Dzyubenko G. Shape preserving approximation of functions

In the thesis a number of classical in form estimates of Shape Preserving Approximation (SPA) of functions by polynomials and splines on a finite interval and on the real axe are proved, the place of each of these results among other achievements in the theory of SPA and the classical approximation theory without restrictions is described, a number of examples is proved to show that it is not possible to improve the indicated estimates (in the sense of order of approximation, etc.), and a brief overview of the topic over the last thirty years is made. "Shape" refers to changes of sign, or monotonicity, or convexity, or q-monotonicity (on an interval/period) of a function, whereas "preservation of the shape" – also of polynomials/splines that approximate this function. That is, in contrast to the classical approximation without restrictions, in SPA, the approaching polynomials /splines do not oscilate arbitrarily, but preserve the specified geometric properties of the function. It is known that it is quite possible to approximate a monotone, or convex, or q-monotone function (q>2) by algebraic polynomials which will preserve its shape (i.e., the Weierstrass theorem on approximation by polynomials is true for SPA). At the same time, in some cases, the degrees (or speeds) of SPA are much "worse" than the best approximations without restrictions, while in the others – they are "almost the same". Also, in certain cases, the classical in form estimations of approximation without restrictions is stored in the SPA – in others no. In the thesis, in particular, these cases have been clarified, that is the results on validity and invalidity of uniform and pointwise estimates of errors of SPA in terms of different moduli of smoothness are presented.