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)

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