Given for an unknown function \(y\), the noisy samples \(\{\widetilde{y}_k: k=1\ldots,n-1\}\) on a set of equispaced points in the interval \((0,1)\), the problem is to estimate the \(k\)th derivative of \(y\). This is done by taking the \(k\)th derivative of a function \(f\) approximating \(y\). It is assumed that \(f\) and its \(k\)th derivative are square integrable and that in the boundary points 0 and 1 of the interval, \(f\) interpolates the exact values of \(y\). The result is obtained by minimizing \[ \Phi(f)={1\over n-1} \sum_{j=1}^{n-1}(\widetilde{y}_j-f(x_j))^2+\alpha \| f^{(k)}\| ^2_{L^2(0,1)}, \] where \(\alpha\) is a regularization parameter and \(f\) satisfies the constraints \(f(0)=y(0)\) and \(f(1)=y(1)\). It is shown that this problem has a unique solution which is a piecewise polynomial that can be obtained by solving a system of \(2kn\) linear equations. An error bound is given for all the derivatives of \(f\) up until the \((k-1)\)th in terms of the grid size and the regularization parameter.
This paper generalizes the method for estimating the first derivative that was used by M. Hanke and O. Scherzer [Am. Math. Mon. 108, No. 6, 512–521 (2001; Zbl 1002.65029)] and by Y. B. Wang, X. Z. Jia and J. Cheng [Inverse Probl. 18, No. 6, 1461–1476 (2002; Zbl 1041.65024)].