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The general subject of the paper is the minimization over space functions with half-plane supports of certain quadratic functionals defined from a two-variable Hermitian-valued measure. A detailed theory of the minimizing functions is developed for the particular situation where the supports are intervals in the lexicographic ordering of the integer plane. The main results are concerned with topics such as stability properties, recurrence relations and spectral factorization, which play a significant role in digital signal processing and estimation theory.

The generalized Schur representation of a function matrix $\Omega ( e^{i\theta } )$ satisfying $\| \Omega \|_\infty \leqq1$ is investigated in connection with certain results concerning the extensions of block-Hankel operators acting on Hilbert spaces. Various properties of such representations are elucidated, including a parametrization of $\Omega ( e^{i\theta } )$ in terms of a double sequence of Schur parameter matrices. Special attention is paid to the way in which the representation and parametrization of the shifted function $e^{ik\theta } \Omega (e^{i\theta } )$ are related to those of $\Omega ( e^{i\theta } )$. In particular, the asymptotic behavior of the shifted representation for $k \to \pm \infty $ is studied in detail. The whole theory is developed so as to be of direct use in the analysis of half-plane block-Toeplitz systems.

This paper contains a study of the limit function $a ( z_1 ,z_2 )$ of the PLSI polynomials relative to a given $H_2 $-function $b( z_1 ,z_2 )$. It is shown that $a ( z_1 ,z_2 )$ is analytic in the unit bicylinder, and that $b ( z_1 ,z_2 )$ admits a canonical weakly inner-strongly outer factorization if and only if $a( z_1 ,z_2 )$ enjoys a well-defined property of stability. The theory is illustrated by a detailed example.

A natural equivalence relation on Hermitian matrices is introduced to analyze the concepts of Toeplitz distance and displacement rank. Two Hermitian block-matrices are said to be equivalent when they are congruent under a block-triangular Toeplitz transformation. The matrices having the smallest Toeplitz distance within a given equivalence class are identified. This minimum distance equals the displacement rank minus twice the block size.Some $\Sigma $-unitary transfer functions $S( z )$ and some Schur-like functions $\Phi ( z )$ are constructed from the Lyapunov relation defining the displacement rank. These functions are used to characterize the equivalence classes and especially their minimum-distance representatives. A canonical factorization of $S( z )$ corresponds to a Schur-like decomposition of $\Phi ( z )$, involving a sequence of generalized Schur parameters $E_k $. The sets of functions $S( z )$, $\Phi ( z )$ and of Schur sequences $( E_k )$ characterizing an equivalence class are described in detail. Some results are obtained concerning the Schur parameters of the inverse of a given matrix.