A numerical differentiation method based on legendre expansion with super order Tikhonov regularization

Abstract The aim of this paper is to develop a method based on Legendre expansion to compute numerical derivatives of a function from its perturbed data. The Tikhonov regularization combined with a new penalty term is used to deal with the ill posed-ness of the problem. It has been shown that the solution process is uniform for various smoothness of functions. Moreover, the convergence rates can be obtained self-adaptively when we choose the regularization parameter by a discrepancy principle. Numerical tests show that the method gives good results.