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Bayesian Kalman filter (BKF) is an important tool in signal processing, communications, control and statistics. This paper briefly reviews the principle of BKF for Gaussian mixture and proposes a new and efficient method for real-time implementation. Moreover, the close relationship between conventional KF and regularization theory in estimation is reviewed. Using this framework, the problem of sampling, smoothing and interpolation can be treated in a unified framework. New results on under-sampling using non-uniform samples will be presented.