Let
{
α
n
}
\{ {\alpha _n}\}
be a sequence of (not necessarily distinct) points in the open unit disk, and let
\[
B
0
=
1
,
B
n
(
z
)
=
∏
m
=
1
n
α
m
¯
|
α
m
|
(
α
m
−
z
)
(
1
−
α
m
¯
z
)
,
n
=
1
,
2
,
…
,
{B_0} = 1,\quad {B_n}(z) = \prod \limits _{m = 1}^n {\frac {{\overline {{\alpha _m}} }} {{|{\alpha _m}|}}\frac {{({\alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,}
\]
(
α
n
¯
|
α
n
|
=
−
1
\frac {{\overline {{\alpha _n}} }} {{|{\alpha _n}|}} = - 1
when
α
n
=
0
{\alpha _n} = 0
). Let
μ
\mu
be a finite (positive) Borel measure on the unit circle, and let
{
φ
n
(
z
)
}
\{ {\varphi _n}(z)\}
be orthonormal functions obtained by orthogonalization of
{
B
n
:
n
=
0
,
1
,
2
,
…
}
\{ {B_n}:n = 0,1,2, \ldots \}
with respect to
μ
\mu
. Boundedness and convergence properties of the reciprocal orthogonal functions
φ
n
∗
(
z
)
=
B
n
(
z
)
φ
n
(
1
/
z
¯
)
¯
\varphi _n^*(z) = {B_n}(z)\overline {{\varphi _n}(1/\overline z )}
and the reproducing kernels
k
n
(
z
,
w
)
=
∑
m
=
0
n
φ
m
(
z
)
φ
m
(
w
)
¯
{k_n}(z,w) = \sum \nolimits _{m = 0}^n {{\varphi _m}(z)\overline {{\varphi _m}(w)} }
are discussed in the situation
|
α
n
|
⩽
R
>
1
|{\alpha _n}| \leqslant R > 1
for all
n
n
, in particular their relationship to the Szegö condition
∫
−
π
π
ln
μ
′
(
θ
)
d
θ
>
−
∞
\int _{ - \pi }^\pi {\ln \mu ’(\theta )d\theta > - \infty }
and noncompleteness in
L
2
(
μ
)
{L_2}(\mu )
of the system
{
φ
n
(
z
)
:
n
=
0
,
1
,
2
,
…
}
\{ {\varphi _n}(z):n = 0,1,2, \ldots \}
. Limit functions of
φ
n
∗
(
z
)
\varphi _n^{\ast }(z)
and
k
n
(
z
,
w
)
{k_n}(z,w)
are obtained. In particular, if a subsequence
{
α
n
(
s
)
}
\{ {\alpha _{n(s)}}\}
converge to
α
\alpha
, then the subsequence
{
φ
n
(
s
)
∗
(
z
)
}
\{ \varphi _{n(s)}^{\ast }(z)\}
converges to
\[
e
i
λ
1
−
|
α
|
2
1
−
α
¯
z
1
σ
μ
(
z
)
,
λ
∈
R
,
{e^{i\lambda }}\frac {{\sqrt {1 - |\alpha {|^2}} }} {{1 - \overline \alpha z}}\frac {1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf {R}},
\]
where
\[
σ
μ
(
z
)
=
2
π
exp
[
1
4
π
∫
−
π
π
e
i
θ
+
z
e
i
θ
−
z
ln
μ
′
(
θ
)
d
θ
]
.
{\sigma _\mu }(z) = \sqrt {2\pi } \exp \left [ {\frac {1} {{4\pi }}\int _{ - \pi }^\pi {\frac {{{e^{i\theta }} + z}} {{{e^{i\theta }} - z}}} \ln \mu ’(\theta )d\theta } \right ].
\]
The kernels
{
k
n
(
z
,
w
)
}
\{ {k_n}(z,w)\}
converge to
1
/
(
1
−
z
w
¯
)
σ
μ
(
z
)
σ
μ
(
w
)
¯
1/(1 - z\overline w ){\sigma _\mu }(z)\overline {{\sigma _\mu }(w)}
. The results generalize corresponding results from the classical Szegö theory (concerned with the polynomial situation
α
n
=
0
{\alpha _n} = 0
for all
n
n
).