Abstract
A modification of the classical theory of spherical harmonics is presented. The unit sphere S in \({\mathbb{R}^3 = \{(x,y,t)\}}\) is replaced by the half-sphere S + in the upper half space, the Euclidean scalar product on S by a non-Euclidean one on S +, and the Laplace equation \({\Delta h = 0}\) by the equation \({t\Delta v + \frac{\partial v }{\partial t}= 0}\). It will be shown that most results from the theory of spherical harmonics in \({\mathbb{R}^3}\) stay valid in this modified setting.
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Dedicated to Thomas Hempfling on the occasion of his 50th birthday.
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Leutwiler, H. Modified Spherical Harmonics. Adv. Appl. Clifford Algebras 27, 1479–1502 (2017). https://doi.org/10.1007/s00006-016-0657-y
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DOI: https://doi.org/10.1007/s00006-016-0657-y