Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations

We consider adaptive estimating the value of a linear functional from indir ect white noise observations. For a flexible approach, the problem is embed ded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collect ion of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functi onals. The case of discretized indirect white noise observations is studied , and the adaptive estimator in this setting is developed.