6]. For additional sources and discussion see the Notes at the end of each of these
chapters. Generalizations to matrix valued Schur functions were considered in ;
for tangential versions, see [23, pp. 97-99] and Section 8 below. In Section 9 we
present a tangential analogue for higher order angular derivatives for matrix valued
The analysis in both Sections 8 and 9 (as well as in a number of other sections)
makes extensive use of the reproducing kernel Hilbert spaces H(S) for p x q matrix
valued Schur functions (see Section 2 below for the precise definition) that were
introduced by de Branges and Rovnyak in  and . The use of these spaces
to study angular derivatives seems to have been initiated in [23, Chapter 8] and,
independently for scalar Schur functions, by Sarason in ; see also his monograph
 for further extensions. Some general classes of interpolation problems for
square mvf's that include constraints on the boundary have also been considered
from another point of view in  and .
The results of Sections 9 and 12 are used in Section 14, where the lossless
inverse scattering problem (LISP) is discussed. This problem may be stated as
Given S £
find all J-inner
G that are analytic in D and meet the
(ip, -s(z))e(z)je(zy ( _£(z) ) o (|*| i).
The LISP was first considered for scalar Schur functions in . Therein necessary
and sufficient conditions for the existence of a rational solution with one or more
poles on the boundary were expressed in terms of the representing measure for the
Caratheodory function2 (1 + S)(l — S)~x. This study was motivated by questions
in network theory and stochastic estimation theory. Some other applications of
boundary interpolation problems are indicated in .
The general LISP for square matrix valued Schur functions was solved in .
There too it proved convenient to work with the (now matrix valued) Caratheodory
function (Ip -f S)(IP — 5) _ 1 . The LISP for general p x q matrix valued Schur
functions was considered in [23, Section 8]. An explicit construction of all rational
solutions which are analytic in the closed unit disk D and of the solutions with one
simple pole on the boundary was given there. In Section 14 we shall extend this
analysis to obtain a description of all the rational solutions of the LISP with an
arbitrary number of poles on the boundary, simple or not. There too conditions for
the existence of a solution to this problem will be formulated directly in terms of
A n x n mvf KLJ(z) defined on ft x Q is said to be a positive kernel if
] T u^K^iuj^Uj 0
The precise definition of a J-inner mvf will be given in Section 4.
See Section 3 for the definition.