In this article we shall use two special classes of reproducing
kernel Hilbert spaces (which originate in the work of de Branges
[dB) and de Branges-Rovnyak [dBRl), respectively) to solve matrix
versions of a number of classical interpolation problems. Enroute
we shall reinterpret de Branges' characterization of the first of
these spaces, when it is finite dimensional, in terms of matrix
equations of the Liapunov and Stein type and shall subsequently
draw some general conclusions on rational m x m matrix valued
functions which are "J unitary" a.e. on either the circle or the
line. We shall also make some connections with the notation of
displacement rank which has been introduced and extensively studied
by Kailath and a number of his colleagues as well as the one used
by Heinig and Rost [HR). The first of the two classes of spaces
alluded to above is distinguished by a reproducing kernel of the
special form K (>.) = J - U(>')JU(w)* (Ll) w Pw(>') , in
which J is a constant m x m signature matrix and U is an m x m J
inner matrix valued function over ~+, where ~+ is equal to either
the open unit disc ID or the open upper half plane (1)+ and
Pw(>') is defined in the table below.
General
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