On determination of the optimal factor of a nonnegative matrix-valued function

Abstract:

Let $F = [f_{ij}]$, $1 < i$, $j \leqq q$, be a measurable, nonnegative definite $q \times q$ matrix-valued function defined on the unit circle $C$. It is known that when $\mathbfit {F}$ and $\log \det \textbf {F}$ are in $L_1(C)$, $\mathbfit {F}$ admits a factorization of the form $F = \mathbf {\Phi } \mathbf {\Phi }^\ast$, where $\mathbf {\Phi }$ is an optimal, full rank function in $L_2^{0+}(C)$. Under the additional assumption that $\{ (\prod \nolimits _{i = 1}^q f_{ii})/\det F\}$ is in $L_1(C)$, an iterative procedure which yields an infinite series for $\mathbf {\Phi }$ in terms of $\mathbfit {F}$ is given. The optimal function $\mathbf {\Phi }$ plays a significant role in the multivariate prediction theory of stochastic processes. The present work generalizes the results of several authors concerning the determination of the optimal factor $\mathbf {\Phi }$.