Abstract
In the present paper, filtering, prediction and smoothing problems are investigated in a unified manner by introducing a function space description. To these estimation problems, a unified expression for the mean square error is first obtained with the help of linear operators. An original problem of finding a linear minimum mean square error estimator is then reduced to a dual problem of finding an optimal element in a Hilbert space that minimizes a quadratic form associated with the mean square error. By solving this optimization problem, a Wiener-Hopf equation in the function space description is derived.
To each of these filtering, prediction and smoothing problems, the corresponding explicit expressions of this dual problem and of a Wiener-Hopf equation in the function space description are obtained. It is shown that a linear optimal regulator is dual to a filtering problem (or to a prediction problem), while a linear optimal regulator with a jump condition on the state variable is dual to a smoothing problem. The basic results of an estimation problem reported earlier are derived in a simple and perspective manner by solving these dual optimization problems.
