Almost Optimal Differentiation Using Noisy Data Original Research Article

We study differentiation of functionsfbased on noisy dataf(ti)+εi. We recoverf(k)either at a single point or on the interval [0, 1] inL2-norm. Under stochastic assumptions onfandεi, we determine the order of the errors of the best linear methods which use n noisy function values. Polynomial interpolation for the pointwise problem and smoothing splines for the problem inL2-norm are shown to be almost optimal. The analysis involves worst case estimates in reproducing kernel Hilbert spaces and a Landau inequality.