Mathematical constructions in optimal linear filtering theory

Abstract

The linear optimal ltering problems in innite dimensional Hilbert spaces and their extensions are investigated. The quality functional is allowed to be a general quadratic functional dened by a possibly degenerate operator. We describe the solution of the stable and the causal ltering problems. In case of the causal ltering we establish the relation with a relaxed causal ltering problem in the extended space. We solve the last problem in continuous and discrete cases and give the necessary and sucient conditions for the solvability of the original causal problem and conditions for the analogue of Bode{Shannon formula to dene an optimal lter.