On estimation of time dependent spatial signal in Gaussian white noise. (Q1766014)

The estimation problem for an observation model of the form \[ Y^\varepsilon(dt,dx)=S(t,x)dtdx+\varepsilon W(dt,dx),\quad t\in [0,T],\;x\in [0,1]^n, \] is considered, where \(\varepsilon \) is a small parameter, \(W\) is a cylindrical orthogonal random measure, and the signal \(S(t,x)\) is a 1-periodic function in each component of \(x\) that obeys appropriate Sobolev smoothness in \(t\). Two approaches to estimation of \(S(t,x)\) are proposed: a combination of the Chentsov projection with a kernel estimator, in which case optimality of the convergence rate for the mean square risk is achieved with a proper choice of parameters, and a symbiosis of the Chentsov projection with Kalman type on-line estimators, which achieves the same rate in \(\varepsilon \) for the mean square risk.