A modification of the classical theory of spherical harmonics in four dimensions is presented. The space \(\mathbb {R}^4 = \{(x,y,t,s)\}\) is replaced by the upper half space \({\mathbb {R}}_{+}^{4}=\left\{ (x,y,t,s), s > 0 \right\} \), and the unit sphere S in \(\mathbb {R}^4\) by the unit half sphere \(S_{+}=\left\{ (x,y,t,s): x^2 +y^2+ t^2+ s^2 =1, s > 0 \right\} \). Instead of the Laplace equation \(\Delta h = 0\) we shall consider the Weinstein equation \(s\Delta u + k \frac{\partial u }{\partial s}= 0\), for \(k \in \mathbb {N}\). The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on \(S_{+}\). It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In addition we shall deduct—with respect to this non-Euclidean scalar product—an orthonormal system of homogeneous polynomials, which satisfies the above Weinstein equation.

中文翻译：

---

四个维度对修改后的球谐函数的贡献

提出了四个方面的经典的球谐函数理论的修改。空格\（\ mathbb {R} ^ 4 = \ {（x，y，t，s）\} \）被上半部空格\（{\ mathbb {R}} _ {+} ^ {4 } = \ left \ {（x，y，t，s），s> 0 \ right \} \），单位球面S在\（\ mathbb {R} ^ 4 \）中由单位半球面\（ S _ {+} = \ left \ {（x，y，t，s）：x ^ 2 + y ^ 2 + t ^ 2 + s ^ 2 = 1，s> 0 \ right \} \）。代替拉普拉斯方程\（\ Delta h = 0 \），我们将考虑Weinstein方程\（s \ Delta u + k \ frac {\ partial u} {\ partial s} = 0 \），对于\（k \在\ mathbb {N} \）中。S函数上的欧氏标量积将被\（S _ {+} \）上的函数替换为非欧几里得。将显示在此修改的设置中，来自球谐理论的所有主要结果仍然有效。此外，对于这个非欧氏标量积，我们将推导出满足上述Weinstein方程的齐次多项式的正交系统。