Kolmogorov widths and unsaturable approximation of function classes, determined by solutions of mathematical physics' equations (Part II. Function of multiple variables)

Background. The article by K. I. Babenko «On some problems of the theory of approximations and numerical analysis» [2] among a number of important problems of calculus mathematics formulates two problems: 1) calculation of Kolmogorov and Babenko widths for $Q_r(\Omega,M)$ class ($Q_r(\Omega,M)$ class consists of functions with continuous derivatives up to $r$ order in $\Omega$ range and derivatives up to $(2r+1)$ order in $\Omega \backslash \partial \Omega$ range, and the magnitude of a derivative of k order $(r<k \leq 2r+1)$ is evaluated by the inequality $\|D^k f\| \leq cl(d(x,\partial \Omega))^{k-r}$, where $d(x,\partial \Omega)$ is a distance from point x to $\partial \Omega$ range border; 2) building of unsaturable methods of function classes approximation. The present study is devoted to calculation of Kolmogorov and Babenko widths of $\overline{Q}^u_{r,\gamma}(\Omega,M)$ and $Q^u_{r,\gamma}(\Omega,M)$ classes of functions with multiple variables, being a generalization of $Q_r(\Omega,M)$ function class; to building of the optimal in methods order approximation of functions of these classes and building of unsaturable algorithms of approximation, the accuracy of which differs from the accuracy of the accurate ones $O(ln^\alpha n)$ by multiplier, where is $n$ a number of functionals used in algorithm building, $\alpha$ is a certain constant. $\overline{Q}^u_{r,\gamma}(\Omega,M)$, $Q^u_{r,\gamma}(\Omega,M)$ function classes possess solutions of elliptical equations, weakly singular, singular and hypersingular integral equations. Materials and methods. Calculation of Kolmogorov width is based on evaluation of Babenko width from the bottom, evaluation of Kolmogorov width from the top and on usage of a lemma establishing a bond between widths. To evaluate Kolmogorov width from the top it is necessary to build local splines that appear to be optimal methods of approximation of $\overline{Q}^u_{r,\gamma}(\Omega,M)$, $Q^u_{r,\gamma}(\Omega,M)$ function classes. Results and conclusions. The author built optimal methods of approximation of $\overline{Q}^u_{r,\gamma}(\Omega,M)$, $Q^u_{r,\gamma}(\Omega,M)$ function classes that may serve as a base of effective numerical methods of solution of elliptical equations, weakly singular, singular and hypersingular integral equations.