Harmonic and Monogenic Functions on Toroidal Domains

Z. Ashtab, J. Morais, R. Michael Porter
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Abstract

A standard technique for producing monogenic functions is to apply the adjoint quaternionic Fueter operator to harmonic functions. We will show that this technique does not give a complete system in \(L^2\) of a solid torus, where toroidal harmonics appear in a natural way. One reason is that this index-increasing operator fails to produce monogenic functions with zero index. Another reason is that the non-trivial topology of the torus requires taking into account a cohomology coefficient associated with monogenic functions, apparently not previously identified because it vanishes for simply connected domains. In this paper, we build a reverse-Appell basis of harmonic functions on the torus expressed in terms of classical toroidal harmonics. This means that the partial derivative of any element of the basis with respect to the axial variable is a constant multiple of another basis element with subindex increased by one. This special basis is used to construct respective bases in the real \(L^2\)-Hilbert spaces of reduced quaternion and quaternion-valued monogenic functions on toroidal domains.

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环形域上的谐函数和单原函数
产生单原函数的一种标准技术是将邻接四元 Fueter 算子应用于谐函数。我们将证明,这种技术并不能在实体环的 \(L^2\) 中给出一个完整的系统,在这个系统中,环状谐波以一种自然的方式出现。原因之一是这种指数递增算子无法产生指数为零的单元函数。另一个原因是,环的非琐碎拓扑要求考虑与单生函数相关的同调系数,而这一系数显然是以前没有发现的,因为它在简单连接域中消失了。在本文中,我们建立了以经典环面谐波表示的环面谐函数的反向-阿佩尔基础。这意味着该基的任何元素相对于轴变量的偏导数都是另一个基元素的常数倍,而另一个基元素的子指数增加了 1。这种特殊的基被用来在实 \(L^2\)-Hilbert 空间中构建环状域上的还原四元数和四元值单原函数的各自基。
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