Quantization of the Gamma Function

Abstract

A class of weight functions for the upper half-plane is introduced which admit a generalization of Fourier analysis. A weighted Hardy space for the upper half-plane is considered, as well as related Hilbert spaces of entire functions. Both the Hardy space and the spaces of entire functions admit dissipative imaginary shifts. The dissipative property implies that the weight function has an analytic extension to a half-plane which contains the upper half-plane and that the ratio of the shifted function to the given function has nonnegative real part in the half-plane. These conditions are satisfied when the weight function is a constant, in which case the Hilbert spaces of entire functions are the Paley-Wiener spaces of Fourier analysis. The simplest nontrivial examples of such weight functions are constructed from the gamma function. The associated Hilbert spaces of entire functions appear in the theory of the Hankel transformation. The spaces have been called Sonine spaces by James and Virginia Rovnyak because they are related to properties of the Hankel transformation, which were discovered in 1880 by N. Sonine. His contributions are often taken as motivation for the Riemann hypothesis because entire functions appear which are related to zeta functions and for which an analogue of the Riemann hypothesis is true. This feature persists in all the Hilbert spaces of entire functions associated with the present generalization of the gamma function. The generalization is seen as a variation on the classical theme of quantization in special function theory.