1. Introduction
In this paper we shall study a number of tangential interpolation problems in
the Schur class o( p x q matrix valued functions that are analytic and contractive
in the open unit disk when a finite number of interpolation constraints are imposed
on the boundary. We shall work within the framework of the augmented Basic
Interpolation Problem (aBIP). An introduction to this problem, which includes an
account of its development from more elementary problems (such as bitangential
versions of the classical Nevanlinna-Pick and Caratheodory-Fejer problems) as well
as other formulations, appears in [27].
In order to describe the aBIP we need to introduce some notation. Let Hf*9
denote the set of CpXg-valued functions with entries in the Hardy space H
2
of the
unit disk P and let H ^ 1 be abbreviated by H^. Similarly, let L§(T) designate the
Hilbert space of measurable and square integrable C^-valued functions with inner
product
(/, 9) = ^ I'" 9(eU)*/(e")d*. (/, 9 G L*{T)).
The space H2 is identified as the closed subspace of L^(T) which consists of all
/»2TT
/ G L2 W whose Fourier coefficients ji ^ \ e~lUf(elt)dt are equal to zero for
n Jo
£ 0. The symbol (H2) stands for the orthogonal complement of HrJ with respect
to the above inner product. More generally, (H^*9) denote the set of CpX9-valued
functions with entries in H^ . The Schur class of CpXQ-valued analytic contractions
in P is denoted by Spxq. In what follows, H ^ 9 will denote the space oipxq mvf's
with entries that are analytic and bounded on P. With every mvf (matrix valued
function) S G
SpXq
we associate the matrix valued Hermitian form [ , ]s
^
g]s
= i
r9{euy
(sih*
~sv) Heu)dt (ii)
which is defined for every choice of h G
L2P X
(T) and g
L2P X
(T) and any
positive integers k and I. This form is nonnegative:
[ft, h]s 0 for all ft e
L{^q)xk(T),
since ( _
5 (
^ t
r
" ^ ^ \ 0 for almost all t G [0; 2TT].
Throughout this paper Ik stands for the identity matrix in
Ckxk,
J denotes
the signature matrix defined by
ip
-1,
and X * is a convenient shorthand for (X*) when X is invertible. We assume
that
M, TV, P G C n x n and C G C(p+?)xn (1.2)
is a given set of matrices satisfying the Lyapunov-Stein identity
M*PM - N*PN = C*JC (1.3)
and that the mvf
G(z) = M-zN (1.4)
1
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