超复积分变换与Paley-Wiener定理

Hypercomplex Integral Transform and Paley-Wiener Theorem

摘要: 本文基于实Paley-Wiener定理，展示了一种不涉及域移动的方法，将实Paley-Wiener定理推广为复Paley-Wiener定理，该方法也同样可以有效地应用于其他傅里叶型变换。通过研究CK延拓的基本性质和运算规则以及Paley-Wiener定理在余维数为1的空间结构进行分析，发现了CK延拓可以保持函数的一些单演性，在此基础上进一步推广出余维数为p的Paley-Wiener定理。

Abstract: Based on the real Paley-Wiener Theorem, this paper presents a method that does not involve domain shifting, which extends the real Paley-Wiener Theorem to the complex Paley-Wiener Theorem. This method can also be effectively applied to other Fourier-type transforms. By investigating the basic properties and operation rules of CK extension, as well as analyzing the Paley-Wiener Theorem in the spatial structure with codimension 1, it is found that CK extension can preserve some monogenicity of functions. On this basis, the Paley-Wiener Theorem with codimension p is further generalized.