西格尔域上的Radon变换

Radon Transform on the Siegel Domain

摘要: 本文基于Cayley变换，将单位球面上的Radon变换推广到Siegel域，建立了适用于全纯函数的新型Radon变换。通过变量代换，成功将球面上的核函数映射为Siegel域上的Radon核，证明了该变换可表示为Hardy空间到由复平面波张成的闭子空间上的正交投影，并给出了显式的积分核函数表达式。进一步推导了该变换的级数展开形式、对偶变换表达式以及函数重构公式，建立了完整的理论框架。

Abstract: Based on the Cayley transform, this paper generalizes the Radon transform from the unit sphere to the Siegel domain, establishing a novel Radon transform tailored for holomorphic functions. Through a change of variables, the kernel function on the sphere is successfully mapped to the Radon kernel on the Siegel domain. It is proven that this transform can be represented as an orthogonal projection from the Hardy space onto a closed subspace spanned by complex plane waves, with its explicit integral kernel provided. Furthermore, we derive the series expansion of the transform, the expression of its dual transform, and the corresponding function reconstruction formula, thereby constructing a comprehensive theoretical framework.