This book consists of six mutually related papers. Each paper deals with certain interpolation problems for rational matrix-valued functions and each uses the realization or state space approach as a main tool. The latter means the following. Let W(\(\lambda)\) be a rational matrix-valued function which is analytic at infinity. Then W(\(\lambda)\) can be represented in form \(W(\lambda)=D+C(\lambda -A)^{-1}B\), where A is a square matrix whose size may be much larger then the size of W(\(\lambda)\). Also, B, C and D are matrices. This representation, which is called a realization, comes from systems and control theory, and it means that W is the transfer function of an input/output system with state operator A, input operator B, output operator C and external operator D. One of the basic interpolation problems dealt with in the present book is the problem to construct a rational matrix function W(\(\lambda)\), regular at infinity, which has prescribed null (zero) and pole data. In terms of realization this problem reduces to the construction, if possible, of matrices A, B, C and D such that the function \(D+C(\lambda -A)^{-1}B\) has the given null and pole data. In what follows a brief description is given of the contents of each of the six papers in the present book.
The first paper “Realization and interpolation of rational matrix functions” (pp. 1-72), by J. A. Ball, I. Gohberg and L. Rodman, serves as a general introduction. It sets the terminology and makes precise the connections hinted at above. It also contains the solution of a tangential Nevanlinna-Pick problem and a Nevanlinna-Pick- Takagi problem.
The second paper “Interpolation problems for rational matrix functions with incomplete data” (pp. 73-108), by I. Gohberg, M. A. Kaashoek and A. C. M. Ran, solves the above mentioned basic problem for the case when the given null and pole data are incomplete. The results are used to derive the formulas for the factors in a Wiener- Hopf factorization from a new point of view.
In the third paper “Regular rational matrix functions with prescribed pole and zero structure” (pp. 109-122), by I. Gohberg and M. A. Kaashoek, the regularity condition at infinity is removed by using an appropriate Möbius transform.
The fourth paper “Inverse spectral problems for regular improper rational matrix functions” (pp. 123-174), by J. A. Ball, N. Cohen and A. C. M. Ran, solves another variant of the basic interpolation problem, also with no restrictions at infinity. As an application the model reduction problem in \(H_{\infty}\)-control theory is solved, both for the discrete time and continuous time settings.
The fifth paper “Unitary rational matrix functions” (pp. 175-222), by D. Alpay and I. Gohberg, develops a realization theory for rational matrix functions that are unitary on the unit circle or on the imaginary line in the framework of a, generally speaking, indefinite inner product. Minimal additive and minimal multiplicative decompositions of unitary rational matrix functions are described in terms of invariant subspaces of state space operators. As a by-product, a new approach to the inertia theorems is obtained.
This paper also prepares the grounds for the sixth paper “Proper contractions and their unitary completions” (pp. 223-248), by I. Gohberg and S. Rubenstein, which concerns an interpolation- completion problem appearing in electrical network theory (Darlington synthesis). Here the first result is a description in terms of realization of rational matrix functions that are proper contractions. The latter means that the value of W(\(\lambda)\) is a contraction for each real \(\lambda\) and a strict contraction for \(\lambda =\infty\). By using this state space description certain minimal unitary completions of proper contractions are obtained and linear fractional decompositions of such functions are characterized.