4. Innovations, Wold Decomposition, and Spectral Factorization

We begin this chapter by reviewing some basic concepts from the theory of dynamic estimation in the classical setup of Wiener and Kolmogorov. The theory leads naturally to considering certain white noise representations of the observation process, which are prototypes of stochastic dynamical systems described in input-output form. These representations were first introduced in geometric terms in the seminal work of H. Wold on stationary processes and prediction theory. Wold’s ideas have been generalized in many directions. One such generalization will be discussed in this chapter and will form the basis of representation theorems which will be used throughout the book. Generalizations of Wold decomposition have become part of functional analysis and have led to a unifying view of certain fundamental problems in operator theory and Hardy spaces. The operator theoretic (and Hardy space) results which stem from this idea can, in a sense, be seen as function-analytic counterparts of results in the theory of stationary processes and in prediction theory. In Sect. 4.6 we take advantage of this conceptual connection to review, in an economical and essentially self-contained way, some basic parts of Hardy space theory that will be needed in various parts of the book.