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A simple solution to linear least-mean-squared-error smoothing, filtering, and prediction problems that is based on series representations for continuous-time random processes is presented. A method for arbitrarily closely approximating correlation functions is employed to obtain approximants that lead to a simple but general closed-form solution for the impulse-response function of the estimating system. The system is asymptotically optimum, and is automatically synthesized with a canonical structure that is convenient for implementation and amenable to adjustment. The method of solution is valid when the observations are restricted to a finite interval and contain a white component.