STATISTICAL INVERSE ESTIMATION IN HILBERT SCALES

The recovery of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regardi ng the smoothness of the signal is available. For greater flexibility the general problem is embedded in an abstract Hilbert scale. In the a pplications Sobolev scales are used. For the construction of estimator s we employ preconditioning along with regularized operator inversion in the appropriate inner product, where the operator is bounded but no t necessarily compact. A lower bound to certain minimax rates is inclu ded? and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate. Examples include errors-in-variabl es (deconvolution) and indirect nonparametric regression. Special inst ances of the latter are estimation of the source term in a differentia l equation and the estimation of the initial state in the heat equatio n.