(ii) Conditions which guarantee a priori the existence, physical realizability, and stability of the optimal filter. - Rudolf E. Kálmán

At present, a nonspecialist might well regard the Wiener-Kolmogorov theory of filtering and prediction [1, 2] as "classical' — in short, a field where the techniques are well established and only minor improvements and generalizations can be expected. That this is not really so can be seen convincingly from recent results of Shinbrot , Stceg , Pugachev [5, 6], and Parzen . Using a variety of methods, these investigators have solved some long-stauding problems in nonstationary filtering and prediction theory. We present here a unified account of our own independent researches during the past two years (which overlap with much of the work [3-71 just mentioned), as well as numerous new results. We, too, use time-domain methods, and obtain major improvements and generalizations of the conventional Wiener theory. In particular, our methods apply without modification to multivariate problems.

(2) The computational aspect. The classical (more accurately, old-fashioned) view is that a mathematical problem is solved if the solution is expressed by a formula. It is not a trivial matter, however, to substitute numbers in a formula. The current literature on the Wiener problem is full of semi-rigorously derived formulas which turn out to be unusable for practical computation when the order of the system becomes even moderately large...

(1) The general form of the solution (see Fig. 1).