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Recursive algorithms are designed for the computation of the optimal linear filter and all types of predictors and smoothers of a signal vector corrupted by a white noise correlated with the signal. These algorithms are derived under both continuous and discrete time formulation of the problem. The only hypothesis imposed is that the correlation functions involved are factorizable kernels. The main contribution of this work with respect to previous studies lies in allowing correlation between the signal and the observation noise, which is useful in applications to feedback control and feedback communications. Moreover, recursive computational formulas are obtained for the error covariances associated with the above estimates.