ALMOST OPTIMAL DIFFERENTIATION USING NOISY DATA

We study differentiation of functions fl based on noisy data f(t(i)) epsilon(i). We recover f((k)) either at a single point or on the inte rval [0, 1] in L(2)-norm. Under stochastic assumptions on f and epsilo n(i), we determine the order of the errors of the best linear methods which use n noisy function values. Polynomial interpolation for the po intwise problem and smoothing splines for the problem in L(2)-norm are shown to be almost optimal. The analysis involves worst case estimate s in reproducing kernel Hilbert spaces and a Landau inequality. (C) 19 96 Academic Press, Inc.