Every homogeneous polynomial \(p\) can be decomposed to

\[ p=\lvert x\rvert^2 p_1 + \cdots + \lvert x\rvert^{2k} p_k \]

where \(k\) is a suitable integer, and \(p_1,\ldots ,p_k\) are solid harmonics [1, Corollary 17.15].

Link Link to heading

• solid harmonic
• proof
• every polynomial has a harmonic restriction on a sphere
• spherical harmonics \(\iff \) solid harmonics

References Link to heading

1. B. Hall, Quantum Theory for Mathematicians. New York, NY: Springer New York, 2013. doi:10.1007/978-1-4614-7116-5