On holomorphic functions on a strip in the complex plane
Abstract
Let
f
be a holomorphic function on the strip
{z∈C:−α<Imz<α},α>0
, belonging to the class
H(α,−α;ϵ)
defined below. It is shown that there exist holomorphic functions
w
1
on
{z∈C:0<Imz<2α}
and
w
2
on
{z∈C:−2α<Imz<2α}
such that
w
1
and
w
2
have boundary values of modulus one on the real axis and satisfy the relation
w
1
(z)=f(z−αi)
w
2
(z−2αi)
and
w
2
(z+2αi)=
f
¯
(z+αi)
w
1
(z)
for
0<Imz<2
, where
f
¯
(z):=
f(
z
¯
)
¯
. This leads to a "polar decomposition"
f(z)=
u
f
(z+αi)
g
f
(z)
of the function
f(z)
, where
u
f
(z+αi)
and
g
f
(z)
are holomorphic functions for
−α<Imz<α
such that
|
u
f
(x)|=1
and
g
f
(x)≥0
a.e. on the real axis. As a byproduct, an operator representation of a
q
-deformed Heisenberg algebra is developed.