A POLAR DECOMPOSITION FOR HOLOMORPHIC FUNCTIONS ON A STRIP

Abstract

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) = 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 < Im z < α, such that [mid ]u_f(x)[mid ] = 1 and g_f(x) [ges ] 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed.