论文标题
超级空间中的cauchy-kovalevskaya扩展和平面波分解
Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace
论文作者
论文摘要
本文的目的是为双轴狄拉克操作员以超空间中的超空间获得广义的CK扩展定理。在经典的通勤情况下，该结果可以写为作用于单个初始功能的某些差分运算符的贝塞尔类型的功率系列。在超空间设置中，新型结构出现在负数超速的情况下。在这些情况下，CK扩展取决于两个初始函数，两个差分运算符的功率系列作用。这些系列不仅是Bessel类型的，而且在Appell多项式方面产生了额外的结构。这种模式也存在于Pizzetti公式的结构中，该公式在差分运算符方面描述了超球的整合。我们通过研究在超球上整合的平面波中的广义ck扩展为平面波的分解来显式。此外，这些结果可用于将cauchy内核在超空间中的分解中分解为单基础波，这对于反转超ra镜转换非常有用。
The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the bi-axial Dirac operator. In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.
