On the Connection Between the Fueter–Sce–Qian Theorem and the Generalized CK-Extension

Abstract

The Fueter–Sce–Qian theorem provides a way of inducing axial monogenic functions in $\mathbb {R}^{m+1}$ from holomorphic intrinsic functions of one complex variable. This result was initially proved by Fueter and Sce for the cases where the dimension m is odd using pointwise differentiation, while the extension to the cases where m is even was proved by Qian using the corresponding Fourier multipliers. In this paper, we provide an alternative description of the Fueter–Sce–Qian theorem in terms of the generalized CK-extension. The latter characterizes axial null solutions of the Cauchy–Riemann operator in $\mathbb {R}^{m+1}$ in terms of their restrictions to the real line. This leads to a one-to-one correspondence between the space of axially monogenic functions in $\mathbb {R}^{m+1}$ and the space of analytic functions of one real variable. We provide explicit expressions for the Fueter–Sce–Qian map in terms of the generalized CK-extension for both cases, m even and m odd. These expressions allow for a plane wave decomposition of the Fueter–Sce–Qian map or, more in particular, a factorization of this mapping in terms of the dual Radon transform. In turn, this decomposition provides a new possibility for extending the Coherent State Transform (CST) to Clifford Analysis. In particular, we construct an axial CST defined through the Fueter–Sce–Qian mapping, and show how it is related to the axial and slice CST’s already studied in the literature.