Extremal representations of functions of matrices and applications to multivariate prediction

Motivated by two seminal results of multivariate prediction theory by Helson and Lowdenslager and by Wiener and Masani we prove extremal representations of functions of matrices and derive their prediction-theoretic consequences. We also sketch a way to obtain matricial inequalities from our results. The main goal of the paper is the computation of the infimum of a set of values of the form $tr(A\mathrm{\Delta }{A}^{\ast })$, where $\mathrm{\Delta }$ is a given non-negative Hermitian $n\times n$ matrix and the choices for $A$ exhauste a certain set of $n\times n$ matrices. In particular, we focus on norm-bounded unit spheres with certain types of properties of unitary invariance, what allows an application of the theory of majorization.