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 En, and a linear interpolation M from Em 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.
Bibliographical noteFunding Information:
through grant NSF MCS