Asymptotic behavior of the best uniform approximations of individual functions by splines (Q1812897)

Let \(W^ r H_ \omega\) (\(r=0,1,\ldots\)) be the class of functions \(f\) that are \(r\)-times continuously differentiable on \([0,2\pi]\) for which \(\omega(f^{(r)},t)\leq \omega(t)\), \(t\in[0,2\pi]\) where \(\omega(f^{(r)},t)=\sup_{| t'-t''|\leq t}| f^{(r)}(t')- f^{(r)}(t'')|\) and \(\omega(t)\) is a given modulus of continuity. The author shows that in this class of functions there exists a mapping for which the error of the best approximation by splines of minimal deficiency asymptotically coincides with the upper bound of approximation of the functions of class \(W^ r H_ \omega\) by these same splines.