Wavelet approach to numerical differentiation of noisy functions
Jianzhong Wang · Communications on Pure & Applied Analysis · 2007
We apply wavelet transform in the study of numerical differentiation for thefunctions which are infected by noise. Because of the presence of noise, theobserved noisy function is not differentiable. In order to estimate thederivatives of the target function from its observation, a pretreatment of theobservation is necessary. The paper introduces differential approximationwavelets (DA-wavelets) so that the DA-wavelet transforms of the observedfunction approximate the derivatives of the target function. The paper alsoshows that the derivatives of compactly supported splines lead to a certaintype of DA-wavelet transforms, which are difference formulas for computingderivatives. The relation between difference formulas and splines enables usto construct various difference formulas via splines and to estimate thecomputing errors of difference formulas in the spline framework.