Tensor spherical harmonics and tensor multipoles. I. Euclidean space

Two bases in the Hilbert space of tensor fields on the unit sphere are discussed: the tensor spherical harmonics and the tensor multipoles. For vector fields these two bases are related by an orthonormal transformation whose coefficients are shown to be Clebsch–Gordan coefficients. This remark suggests a method of building multipole bases for higher order tensor fields. The second order tensor multipoles are studied in detail as well as their relations with the symmetric ones defined by several authors for application to the gravitational radiation.