Asymptotically efficient estimation of analytic functions in Gaussian noise

Abstract

The problem of recovery of an unknown regression function f(x), x∈R¹, from noisy data is considered. The function f(.) is assumed to belong to a class of functions analytic in a strip of the complex plane around the real axis. The performance of an estimator is measured either by its deviation at a fixed point, or by its maximal error in the L_∞-norm over a bounded interval. It is shown that in the case of equidistant observations, with an increasing design density, asymptotically minimax estimators of the unknown regression function can be found within the class of linear estimators. Such best linear estimators are explicitly obtained.