Approximation of some classes of functions by local splines (Q1569311)

In this paper the Kolmogorov and Babenko widths of the class of functions \(Q_{r{\gamma}p}(\Omega,M)\) are evaluated. The class \(Q_{r{\gamma}p}(\Omega,M)\) consists of functions whose derivatives increase unboundedly as the boundary of the domain \(\Omega\) is approached. Here the positive numbers \(r,{\gamma},p,M\) are the growth order of function derivatives. Local splines whose errors are of the same order as the evaluated widths are constructed. The estimates for \(d_n(Q_{r{\gamma}p}(\Omega,M),L_q)\) have been obtained, where \(d_n(X,B)\) being the Kolmogorov width is defined as \[ d_n(X,B)=\inf_{L^n \in{B}} \sup_{x\in{X}} \inf_{u\in{L^n}} \|{x-u}\|, \] where \(B\) is a Banach space, \(X\subset{B}\) a compact set, and \(L^n\) the set of \(n\)-dimensional subspaces of the linear space \(B\).