Direct Estimation of Linear Functionals from Indirect Noisy Observations

The authors study the efficiency of the linear–functional strategy, as introduced by Anderssen (1986), for inverse problems with observations blurred by Gaussian white noise with known intensity δ. The optimal accuracy is presented and it is shown how this can be achieved by a linear–functional strategy based on the noisy observations. This optimal linear–functional strategy is obtained from Tikhonov regularization of some dual problem. Next, the situation is treated when only a finite number of noisy observations, given beforehand is available. Under appropriate smoothness assumptions best possible accuracy still can be attained if the number of observations corresponds to the noise intensity in a proper way. It is also shown that, at least asymptotically this number of observations cannot be reduced.