This paper discusses the fixed-point smoothing and filtering problems given lumped covariance function of a scalar signal process observed with additive white Gaussian noise. The recursive Wiener smoother and filter are derived by applying an invariant imbedding method to the Volterra-type integral equation of the second kind in linear least-squares estimation problems. The resultant estimators in Theorem 2 require the information of the crossvariance function of the state variable with the observed value, the system matrix, the observation vector, the variance of the observation noise and the observed value. Here, it is assumed that the signal process is generated by the state-space model. The spectral factorization problem is also considered in Sects. 1 and 2.