Optimal Approximation of Linear Operators Based on Noisy Data on Functionals Original Research Article

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{σ21, ..., σ2n}. The a priori measure μ 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, σi′s, and the eigenvalues of the correlation operator of the a priori distribution v = μS−1 on G.