10 ON BOUNDARY INTERPOLATION
If SeS
p x g
, then
M
,
) =
J . - 5 ( « W - „,
is a positive kernel on D x D. The corresponding reproducing kernel Hilbert space
will be referred as to H(S).
The following alternate characterization of H(S), as the space of all vector
functions / G H2 such that
«(/):= sup{||/ + S^p-||^j } (2.6)
is finite and ||/||^(s) ^(/)» is due to de Branges and Rovnyak [17], [18].
The next lemma expresses the reproducing kernel Au of the space H(S) in
terms of the nonnegative form [ , ]s defined via (1.1).
LEMMA 2.8. The formula
w'l=K*r)?(*r)£
"
is valid for every pair of points z and uo in D.
PROOF.
It is readily seen that the functions
,M-{«.r)m -1 9
"
( 0
= - p F
(28)
belong to Hpf 9 and (H^^) - 1 , respectively. Thus, as —j—r is the reproducing
kernel for H??, it follows from definition (1.1) that
lfuV' fzX]s =
\( -s* iq){ s&y ) h V s{ly J fx/L^m
= ( A ^ , ) + (&,y, ^ ) = z*Aw(z)y,
\ P « / H ; \ P* /LJ(T )
for every choice of x and y in
Cp
and hence that (2.7) is valid.
The following simple observation will be useful.
LEMMA
2.9. Let [, ]s be the form defined in (1.1) and let F^iC) be a (p + q) x n
mvf defined for £ and u in D. Then the kernel
FM) := [Fu, Fz]s
is positive on D.
PROOF. For every choice of an integer r and of points CJI, ..., UJT G P,
- s j f ^")-(_sfh-
"T))p(e",ft-
where
% ) = (F
W l
(C),...,F
W r
(0).
Thus, the matrix on the left hand side of the second line of the proof is positive
semidefinite. Therefore, FLJ(z) : 0.
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