Optimal numerical approximation of a linear operator

Let S be a bounded linear transformation from a. Hilbert space B to a Hilbert space Σ. Then Su can be thought of as the solution of a linear differential equation with right-hand side, initial data, or boundary data u. Given the incomplete information Nu = v, ∥u∥_B ≤ 1 about the data, where N is a linear operator from B to a Euclidean space E_n, and a linear interpolation M from E_m to Σ, one defines the optimal approximation to Su to be the point Mâ(v) in the range of M which is the center of the smallest ball containing all points of the form Su with Nu = v and ∥u∥_B ≤ 1 and centered in M. A characterization is given for the optimal approximation Mâ(v). It is shown to be unique and, in general, nonlinear. Simpler approximations and relations with other concepts of optimality are investigated.