Prymak A. Shape-preserving approximation by splines with fixed knots.

Investigation of shape preserving approximation by splines is presented in the thesis. General theorem about reducing of 3-monotone uniform approximation of 3-monotone function by splines to those of local convex approximation of its derivative in L1 norm, with interpolation at the knots. As the corollaries, new Jackson type estimates are obtained for 3-monotone approximation by splines with the equidistant and Chebyshev knots. Constructions for smoothing of such splines are developed, the splines of minimal defect of 3-rd and 4-th degree are found. Negative results for s-monotone approximation by splines with fixed knots and polynomials, for s>3. The constructions of “bad” functions for shape preserving approximation are independent of number of the knots of approximating spline and of degree of approximating polynomial.