Definition. A Schur function
is defined as a complex function analytic and of modulus not exceeding
unity in
.
Property. The function
Proof.
Suppose
is positive real. Then for
,
re
re
is PR. Consequently,
is minimum phase which implies all roots of
lie in the unit circle.
Thus
is analytic in
. Also,
By the maximum modulus theorem,
Conversely, suppose
is Schur. Solving Eq.(C.84) for
and taking the real part on the unit circle yields
If
is constant, then
is PR. If
is not
constant, then by the maximum principle,
for
. By
Rouche's theorem applied on a circle of radius
,
, on
which
, the function
has the same number of
zeros as the function
in
. Hence,
is
minimum phase which implies
is analytic for
. Thus
is PR.