On approximate recursive prediction of stationary stochastic processes

A new approach to the problem of designing a low-dimensional recursive filter for a stationary process is presented. This problem, which is equivalent to a weighted L 2 approximation problem by rational functions of preassigned degree k, is solved in a different metric induced by the corresponding Hankel matrices. The filters so obtained satisfy, in contrast to existing solutions, certain desirable properties. In particular their performances admit a priori bounds. These involve the performance of the optimal filter, the L ¥ norm of the power spectrum and the singular values sk of the Hankel matrix corresponding to the optimal Wiener filter. The speed of decrease of the sk is then characterized in terms of the spectral density of the process and its past-future canonical correlation coefficients.