Kolmogorov diameters and unsaturable methods of approximation of functionclasses, determined by solutions of mathematical physics' equations (Part I. Function of single variable)

Background. Among the important problems of calculus mathematics there can be formulated two problems: calculation of Kolmogorov and Babenko diameters in $Q_r(\Omega, M)$ class; development of unsaturable methods of function compacts approximation. The author calculated Kolmogorov and Babenko diameters of $\overline{Q}^u_{r,\gamma}(\Omega, M)$ and $Q^u_{r,\gamma}(\Omega, M)$ function classes, being the generalization of the function class; built optimal in method order approximations of the said classes; built unsaturable algorithms of the said classes' approximation. Accuracy of the unsaturable algorithms differs from optimal $O(ln^\alpha n)$ multipliers, where $n$ - number of functionals used in algorithm construction, $\alpha$- certain constant. $\overline{Q}^u_{r,\gamma}(\Omega, M)$ and $Q^u_{r,\gamma}(\Omega, M)$ function classes include solutions of elliptic equations, weakly singular, singular and hypersingular integral equations. Materials and methods. Calculation of Kolmogorov diameter is based on the estimate at the bottom of Babenko diameter, on the estimate on the top of Kolmogorov diameter and on usage of the lemma connecting the said two diameters. To estimate the top of Kolmogorov diameter one builds local splines, which appear to be the optimal methods of approxaimation of $\overline{Q}^u_{r,\gamma}(\Omega, M)$ and $Q^u_{r,\gamma}(\Omega, M)$ function classes. Results. The author developed optimal methods of approxaimation of $\overline{Q}^u_{r,\gamma}(\Omega, M)$ and $Q^u_{r,\gamma}(\Omega, M)$, function classes, which may form a base of effective numerical methods of solution of elliptic equations, weakly singular, singular and hypersingular integral equations. Conclusions. The splines, built in the work, may form a base for construction of effective numerical methods of solutions of elliptic equations, weakly singular, singular and hyper singular integral equations.