On adaptive inverse estimation of linear functional in Hilbert scales

We address the problem of estimating the value of a linear functional 〈f, x〉 from random noisy observations of y = 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.