Optimal Approximation of Linear Operators Based on Noisy Data on Functionals

For a linear operator S: F ? G, where F is a Banach space and G is a Hilbert space, we pose and solve the problem of approximating elements g = Sf, f ? F, based on noisy values of n linear functionals at f. The noise is assumed to be Gaussian with correlation matrix D = diag{?21, ..., ?2n}. The a priori measure ? on F is also Gaussian. We show how to choose the functionals from a ball to minimize the expected error of approximation. The error of the optimal approximation is given in terms of n, ?i?s, and the eigenvalues of the correlation operator of the a priori distribution v = ?S?1 on G.