Let f be a holomorphic function on the strip {z
∈ [Copf ] : −α < Im z < α},
where α > 0, belonging to the class [Hscr ](α,−α;ε)
defined below. It is shown that there exist holomorphic functions
w1 on {z ∈ [Copf ] : 0 < Im
z < 2α} and w2 on {z
∈ [Copf ] : −2α < Im z < 2α},
such that w1 and w2 have boundary
values of modulus one on the real axis, and satisfy the relations
w1(z)=f(z-αi)w2(z-2αi) and w2(z+2αi)=f(z+αi)w1(z)
for 0 < Im z < 2α, where f(z)
:= f(z). This leads to a ‘polar
decomposition’ f(z) = uf(z
+ αi)gf(z) of the
function f(z), where uf
(z + αi) and gf(z)
are holomorphic functions for −α < Im z <
α, such that [mid ]uf(x)[mid ] = 1
and gf(x) [ges ] 0 almost everywhere on the real axis. As a
byproduct, an operator representation of a q-deformed Heisenberg
algebra is developed.