Let s be the periodic spline of degree 2r-1 on a finite uniform mesh which agrees with \(f\in C^{2r}\) at the knots. Then both \(\| s- f\|_{\infty}\) and \(\| s'-f'\|_{\infty}\) are \(O(h^{2r})\) but the error in approximations to higher derivatives worsens as the order of derivative increases. This situation can be remedied by the use of iterated splines. For example the first iterated spline is constructed by fitting the periodic splines of degree 2r-1 to the first derivative of the original spline at the knots. The first derivative of this spline at the knots approximates the second derivate of the original function again with error \(O(h^{2r})\). The authors investigate the general problem, and show that with the use of iterated splines the approximation of higher derivatives at the knots can be found with error \(O(h^{2r})\).