Hermite spectral and pseudospectral methods for numerical differentiation

A numerical differentiation problem for a given function with noisy data is discussed in this paper. A mollification method based on spanned by Hermite functions is proposed and the mollification parameter is chosen by a discrepancy principle. The convergence estimates of the derivatives are obtained. To get a practical approach, we also derive corresponding results for pseudospectral (Hermite-Gauss interpolation) approximations. Numerical examples are given to show the efficiency of the method.