论文标题
关于Fueter-Sce-Qian定理与广义CK-Extension之间的连接
On the connection between the Fueter-Sce-Qian theorem and the generalized CK-extension
论文作者
论文摘要
Fueter-SCE-QIAN定理提供了一种从一个复杂变量的Holomorphic固有函数中诱导轴向单基因函数诱导轴向单基因函数。 Fueter和SCE最初证明了该尺寸$ m $使用点式差异的情况，而Qian使用相应的傅立叶乘数证明了$ m $的延伸。 In this paper, we provide an alternative description of the Fueter-Sce-Qian theorem in terms of the generalized CK-extension.后者以$ \ mathbb {r}^{m+1} $在对实际行的限制方面表征了cauchy-riemann运算符的轴向零解决方案。这导致在$ \ mathbb {r}^{m+1} $中轴向单基函数的空间与一个真实变量的分析函数的空间之间的一对一对应关系。 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.这些表达式允许Fueter-SCE-QIAN地图的平面波分解，或者更多地是根据双rad radton变换对该映射的分解。 In turn, this decomposition provides a new possibility for extending the Coherent State Transform (CST) to Clifford Analysis.特别是，我们构建了通过Fueter-Sce-Qian映射定义的轴向CST，并展示了它与文献中已经研究的轴向和切片CST的相关性。
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 CSTs already studied in the literature.
