A Polar Decomposition for Holomorphic Functions on a Strip

Let f be a holomorphic function on the strip { z ∈ C : −α < Im z < α}, where 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 < Im z < 2α} and w 2 on { z ∈ C : −2α < Im z < 2α}, such that w 1 and w 2 have boundary values of modulus one on the real axis, and satisfy the relationsw 1 ( z ) = f ( z − 2 α i ) w 2 ( z + 2 α i )   and   w 2 ( z + 2 α i ) = f ¯ ( z + α i ) w 1 ( z )for 0 < Im z < 2α, where f ¯ ( z ) : = f ( z ¯ ) ¯ . This leads to a ‘polar decomposition’ f ( z ) = u f ( z + α i ) g f ( z ) of the function f ( z ), where u f ( z + α i ) and g f ( z ) are holomorphic functions for −α < Im z < α, such that ∣ u f ( x )∣ = 1 and g f ( x ) ⩾ 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q ‐deformed Heisenberg algebra is developed.