Numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform

In this article, we discuss a classical ill-posed problem on numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform (DCT). After implementing an eigenvalue decomposition of the second-order difference matrix in the penalty we obtain an explicit minimizer of the Tikhonov functional in a linear combination of DCT-2 components. The choice of the regularization parameter thus can be properly chosen after calling a single variable bounded nonlinear function minimization. Moreover, we propose two approaches to weaken the Gibbs phenomena appearing near the boundary of the reconstructed derivatives. Several numerical examples are provided to show the computational efficiency of our proposed algorithm. Finally, the extension to the multidimensional numerical differentiation leads the approach to a wider scope.