Stable Numerical Differentiation Algorithms Based on the Fourier Transform in Frequency Domain

A class of stable numerical differential algorithms is constructed based on the Fourier transform. The instability of the numerical differentiation problem is overcome by modifying the integral “kernel” in the frequency domain. The convergence of the approximate derivatives is ensured based on some reasonable assumptions of the modified “kernel” function. The a-posteriori choice strategy of the regularization parameter is considered. Moreover, the convergence analysis and error estimate of the approximate derivatives are also given.