2 ON BOUNDARY INTERPOLATION
is regular:
det G(z) = det (M - zN) 0. (1.5)
The partition
C = (
S!1
V Ci G C
pxn
, C2 G
C9Xn,
(1.6)
which enables us to express the Lyapunov-Stein equation (1.3) in the form
M*PM - N*PN = C\Cx - CIC2% (1.7)
will be useful. The symbol aBIP(M, AT, P, C) will be used to denote the following
interpolation problem (which is a relaxed version of the aBIP that will be discussed
below):
(1) Find necessary and sufficient conditions which insure the existence of a
Schur function S G Spxq such that
{-sh T
)C G ( O"€(w'"")0 (L8)
and
Ps P, (1.9)
where
pS: = s f
G
^
, c ,
O r
~T})CG{eU)~ldt
= [CG{Q-\ CG(C)-
1
]
5
. (1.10)
(2) Describe the set S(M, N, P, C) o/ a// swcft rav/'s.
If the mvf G(z) is not invertible everywhere on T, the condition (1.9) is meant in
the following sense: the integral
h I"
G{e%r*c*
(-sfrr ~ V )
CG{e"rldt (L11)
converges to a matrix Ps which satisfies inequality (1.9).
The aBIP(M, N, P, C) can be formulated without mention of the space H ^
with the help of the symbol
H(z) = zM* - N* (1.12)
and the following remark:
REMARK
1.1. Condition (1.8) is equivalent to the following two conditions
B(C) := ( IP, -5(C) ) CG(C)-1 G H f " (1.13)
and
B(Q := F(C)
_1
C* (
~Sr^
) G
H^x?.
(1.14)
Lq
PROOF. It is easily seen that a function /(£) belongs to H2 if and only if the
function C/(0* belongs to H^-. Therefore,
ff(0_1C* ( ~^(C) ) G H£x' C ( -5(C)*, /, ) CH(0~* G ( H f V
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