Average case optimal algorithms in Hilbert spaces

We study optimal algorithms and optimal information for an average case model. This is done for linear problems in a separable Hilbert space equipped with a probability measure. We show, in particular, that for any measure a (linear) spline algorithm is optimal among linear algorithms. The spline algorithm is defined in terms of the covariance operator of the measure. We provide a condition on the measure which guarantees that the spline algorithm is optimal among all algorithms. The problem of optimal information is also solved.