Optimal approximation of linear operators based on noisy data on functionals (Q1801571)

Let \(S\) be a linear continuous operator acting from a Banach space \(F\) to a Hilbert space \(G\), both separable over the real field. Then the author poses and solves the problem of approximating elements \(g=Sf\), \(f \in F\), based on noisy values of \(n\) linear functionals at \(f\). The noise is assumed to be Gaussian with correlation matrix \(D=\text{diag} \{\sigma^ 2_ 1, \dots,\sigma^ 2_ n\}\). The prior measure \(\mu\) on \(F\) is also Gaussian. The author shows how to choose the functionals from a ball to minimize the expected error of approximation. The error of optimal approximation is given in terms of \(n\), \(\sigma_ i\)'s and the eigenvalues of the correlation operator of the prior distribution \(\nu=\mu S^{-1}\) on \(G\).