On adaptive inverse estimation of linear functionals in Hilbert scales

We address the problem of estimating the value of a linear functional h f , xi from random noisy observations of y 1⁄4 Ax in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of x, of f , and of the noise covariance operator. It is shown that accuracy of this adaptive estimator is worse only by a logarithmic factor than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.