Numerical differentiation by a Fourier extension method with super-order regularization.

Based on the idea of Fourier extension, we develop a new method for numerical differentiation. The Tikhonov regularization method with a super-order penalty term is presented to deal with the illposdness of the problem and the regularization parameter can be chosen by a discrepancy principle. For various smooth conditions, the solution process of the new method is uniform and order optimal error bounds can be obtained. Numerical experiments are also presented to illustrate the effectiveness of the proposed method.