On best uniform approximation by splines in the presence of restrictions on their derivatives (Q1181613)

The author proves a theorem on best approximation of \(2\pi\)-periodic functions \(f\in W_ \infty^ r= \{f\): \(\| f^{(r)}\|_ \infty\leq 1\}\) by \(S_{2n,r}\), the space of \(2\pi\)-periodic splines of degree \(r\) with \(2n\) equidistant nodes. For a positive nonincreasing sequence \(\{\varepsilon_ n\}\) let \[ E:=\sup \{\inf \{\| f-s\|_ \infty:\;s\in S_{2n,r}\cup (1+\varepsilon_ n) W_ \infty^ r\}:\;f\in W_ \infty^ r\}. \] Then it is proved that \(E\asymp n^{-2}\) for \(\varepsilon_ n n^ 2= O(1)\) and \(E\asymp n^{-r} \varepsilon_ n^{1-r/2}\) for \(\varepsilon_ n n^ 2\to\infty\), respectively. This interesting result solves a problem on the asymptotic behavior of certain related \(n\)-width posed by S. B. Steckin.