In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace
H
2
of the simple, unweighted holomorphic Hilbert space
L
2
of functions square-integrable over the surface of the unit disc
{
z
:
|
z
|
<
1
}
of the complex plane, along with a form of the orthogonal projection from
L
2
to
H
2
.
Wirtinger's paper contains the following theorem presented also in Joseph L. Walsh's well-known monograph (p. 150) with a different proof. If
F
(
z
)
is of the class
L
2
on
|
z
|
<
1
, i.e.
∬
|
z
|
<
1
|
F
(
z
)
|
2
d
S
<
+
∞
,
where
d
S
is the area element, then the unique function
f
(
z
)
of the holomorphic subclass
H
2
⊂
L
2
, such that
∬
|
z
|
<
1
|
F
(
z
)
−
f
(
z
)
|
2
d
S
is least, is given by
f
(
z
)
=
1
π
∬
|
ζ
|
<
1
F
(
ζ
)
d
S
(
1
−
ζ
¯
z
)
2
,
|
z
|
<
1.
The last formula gives a form for the orthogonal projection from
L
2
to
H
2
. Besides, replacement of
F
(
ζ
)
by
f
(
ζ
)
makes it Wirtinger's representation for all
f
(
z
)
∈
H
2
. This is an analog of the well-known Cauchy integral formula with the square of the Cauchy kernel. Later, after the 1950s, a degree of the Cauchy kernel was called reproducing kernel, and the notation
A
0
2
became common for the class
H
2
.
In 1948 Mkhitar Djrbashian extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces
A
α
2
of functions
f
(
z
)
holomorphic in
|
z
|
<
1
, which satisfy the condition
∥
f
∥
A
α
2
=
{
1
π
∬
|
z
|
<
1
|
f
(
z
)
|
2
(
1
−
|
z
|
2
)
α
−
1
d
S
}
1
/
2
<
+
∞
for some
α
∈
(
0
,
+
∞
)
,
and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted
A
ω
2
spaces of functions holomorphic in
|
z
|
<
1
and similar spaces of entire functions, the unions of which respectively coincide with all functions holomorphic in
|
z
|
<
1
and the whole set of entire functions can be seen in.
