Optimal average case estimation in Hilbert norms

In contrast to the worst case approach, the average case setting provides l ess conservative insight into the quality of estimation algorithms. In this paper we consider two local average case error measures of algorithms base d on noisy information, in Hilbert norms in the problem element and informa tion spaces. We define the optimal algorithm and provide formulas for its t wo local errors, which explicitly exhibit the influence of factors such as information, information (measurement) errors, norms in the considered spac es, a subset where approximations are allowed, and "unmodeled dynamics." Ba sed on the error expression, we formulate in algebraic language the problem of selecting the optimal approximating subspace. The solution is given alo ng with the specific formula for the error, which depends on the eigenvalue s of a certain matrix defined by information and norms under consideration.