2. PRELIMINARIES
7
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 [39];
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
Schur functions.
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 [17] and [18]. 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 [48]; see also his monograph
[49] 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 [12] and [14].
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
follows:
Given S £
SpXq,
find all J-inner
mvf's1
G that are analytic in D and meet the
constraint
(ip, -s(z))e(z)je(zy ( _£(z) ) o (|*| i).
The LISP was first considered for scalar Schur functions in [20]. 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 [30].
The general LISP for square matrix valued Schur functions was solved in [3].
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
S.
2. Preliminaries
A n x n mvf KLJ(z) defined on ft x Q is said to be a positive kernel if
r
] 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.
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