Data on the sphere, like the CMB temperature fluctuations, are usually analyzed in terms of spherical harmonics expansion. However, hints for anisotropy and/or non gaussianity effects have recently been reported in the CMB data and are, as well, predicted by some cosmological scenarios. Such effects thus deserve to be explored, in the present and future CMB data, and it has been suggested that harmonic analysis could not be the best tool in this purpose. Among other approaches, the multipole vector decomposition, recently introduced, exhibits a friendly behavior under the spatial rotations. For this reason, it is a good candidate for the analysis of such non standard effects. First applications to the CMB data have given encouraging results in this respect.
Here I propose an interpretation of the multipole vector decomposition, in relation with the harmonic analysis. I show that it is a consequence of the Maxwell multipole representation, and that it may be expressed in terms of the harmonic projection. This provides clear and short proofs of the decomposition theorem, as well as a way to go from multipole vectors to harmonic coefficients. A decomposition recently proposed is proven to be unstable, and an other one is proposed, which is stable. The extension to complex polynomials is discussed.