  We show that the multipole vector decomposition, recently introduced by Copi
et al., is a consequence of Sylvester's theorem, and corresponds to the
Maxwell's representation. Analyzing it in terms of harmonic polynomials, we
show that this decomposition results in fact from the application of the
harmonic projection operator and its inverse. We derive the coefficients of the
usual harmonic decomposition from the multipole vectors. We answer to '' an
open question '', first raised by Copi et al., and reported by Katz and Weeks,
by showing that the decomposition resulting from their corollary is unstable.
We propose however a new decomposition which is stable. We generalize these
results to complex functions and polynomials.
