Approximation of Derivatives by the Derivatives of Interpolating Splines

Let $s_{r-1, 2n} (f, x)$ be a spline of degree $r-1$ of defect 1 with $2n$ equidistant nodes which interpolates a function $f$ at the nodes when $r-1$ is odd and at the midpoints of the intervals connecting neighboring nodes when $r-1$ is even. It is known that such splines provide the best approximations of the classes $W^r$ of $2 \pi$-periodic differentiable functions. Moreover, the derivatives $s_{r-1, 2n}' (f, x)$ provide the best approximations of the class of derivatives $f'(x)$ of the functions $f\in W^r$. In this paper, we consider a similar problem on the approximation of derivatives of order $r-1$ and obtain an estimate that is uniform in $r$ and $n$.