Wavelet approach to numerical differentiation of noisy functions

We apply wavelet transform in the study of numerical differentiation for the functions which are infected by noise. Because of the presence of noise, the observed noisy function is not differentiable. In order to estimate the derivatives of the target function from its observation, a pretreatment of the observation is necessary. The paper introduces differential approximation wavelets (DA-wavelets) so that the DA-wavelet transforms of the observed function approximate the derivatives of the target function. The paper also shows that the derivatives of compactly supported splines lead to a certain type of DA-wavelet transforms, which are difference formulas for computing derivatives. The relation between difference formulas and splines enables us to construct various difference formulas via splines and to estimate the computing errors of difference formulas in the spline framework.