Hermite-Clifford分析中的Cauchy-Kovalevskaya扩张定理
The Cauchy-Kovalevskaya Extension Theorem in Hermitean Clifford Analysis
摘要：
厄米特Clifford分析是一种高维函数理论，它以两个厄米特共轭复Dirac算子的同时零解为中心，称为厄米特单演函数。作为构造Hermite单演多项式的正交基的必要步骤，本文建立了这类多项式的Cauchy-Kovalevskaya扩张定理。确定了获得唯一的厄米特单演扩张所需的初始多项式的最小数目，以及它们必须满足的相容性条件。柯西-科瓦列夫斯卡娅延拓原理则允许对球面厄米单演空间进行维数分析，即齐次厄米单演多项式。对于特定的实解析函数，也得到了这个延拓定理的一个版本.
英文摘要：
Hermitean Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitean monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitean monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitean monogenics, i.e. homogeneous Hermitean monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.
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