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.