Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-04T19:17:12.617Z Has data issue: false hasContentIssue false

A POLAR DECOMPOSITION FOR HOLOMORPHIC FUNCTIONS ON A STRIP

Published online by Cambridge University Press:  14 June 2001

KONRAD SCHMÜDGEN
Affiliation:
Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany; [email protected]
Get access

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(zi)w2(z-2αi) and w2(z+2αi)=f(zi)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.

Type
NOTES AND PAPERS
Copyright
© The London Mathematical Society 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)