Skorokhodov D. Optimal recovery of operators and functionals and related extremal problems of Approximation Theory

The thesis is devoted to investigation of classical problems in Approximation Theory on calculating the widths of functional classes, optimization of quadrature formulas on classes of univariate and multivariate functions, the best approximation of multivariate functions by splines, the best recovery of operators and functionals, obtaining sharp Kolmogorov type inequalities for the norms of derivatives. In the first chapter we consider the problem of finding linear widths of classes H^ω of functions defined on the interval [0,1] and having given majorant ω for its modulus of continuity in the space of continuous functions. Kolmogorov widths of these functional classes were found in 1960-1970’s but exact values of their linear widths remain unknown. We find exact values of the first order linear widths of classes H^ω and its periodic analogues in the space of continuous functions. This allows establishing new upper estimates for higher order linear widths of classes H^ω that improve known estimates. We show that these estimates are sharp on a wide class of linear methods – positive minihedral methods, define and calculate new approximative characteristics that is close to relative widths. In the second chapter we consider the problem of optimization of quadrature formulas. In 1980’s it was proved that the rectangle formula is the best quadrature formula on convolution classes K*F of variation diminishing kernels K with rearrangement invariant sets F of periodic univariate functions. Note that convolution classes generalize many important functional classes, e.g. Sobolev classes. We extend this result onto the problem of optimization of interval quadrature formulas and prove optimality of interval rectangle formula – the formula with equal coefficients and equidistant centers of node intervals – on classes K*F. The key part in establishing our result is played by new variation diminishing property of the Steklov kernel on a narrow class of functions that can be represented as the difference of asymmetric perfect splines of zero order. Also, we solve the problem of optimization of quadrature formulas that use as information about the integrand functions the averages over intersections of its domain with hyperplanes of given dimension on the classes of multivariate functions defined by either the majorant for its modulus of continuity or the limitations on the norms of partial derivatives. In the third chapter we study the problem of finding the asymptotic behavior of the best approximation of multivariate functions by splines. We establish sharp asymptotics of the error of the best nonlinear (α,β)−asymmetric approximation of convex bivariate functions by linear continuous splines in terms of the number of elements of triangulations. Study of asymmetric approximations allows us to consider regular and one-sided approximations under one perspective. Important step in proving this result was to solve extremal problem of the best asymmetric approximation of a positive definite quadratic form by linear functions on simplices. Also, we consider the problem of transfinite interpolation of multivariate functions by harmonic splines. We find exact order of asymptotic behavior of the best interpolation of the class W_∞^Δ (Ω) in L_q-metric by harmonic splines and prove that this order does not depend on the dimensionality of the space where the functions are defined. In the fourth chapter we investigate the problem of optimal recovery of operators given exact or non-exact information. We find the error of optimal recovery of class W_∞^2 (G) of multivariate functions defined on a convex body G⊂R^d and having uniformly bounded second order directional derivatives given the values of functions and its gradients in a fixed finite system of points. Also, we solve the problem of optimal recovery of integral operators with non-negative kernels and sums of such operators on classes of functions defined on compacts of metric spaces and having a given majorant for its modulus of continuity given non-exact information on the values of functions in a fixed finite system of points. We devote the fifth chapter to Kolmogorov type inequalities and related problem of the best approximation of operators by linear bounded ones. We find a new sharp Kolmogorov type inequality estimating L_∞-norm of the Marchaud fractional derivative of a function defined on non-negative half-line in terms of L_∞-norm of the function itself and L_s-norm of the second order function derivative. We obtain new sharp Kolmogorov type inequalities for the norms of derivatives of absolutely monotone and multiply monotone functions defined on a finite interval. In addition, we solve the problem of the best approximation of first and second order differentiation operators on the classes of functions defined on a finite interval and having either second order derivative bounded in the space L_p or the Orlicz space or third order derivative bounded in the space L_∞.