  Let $h_g^\infty$ be the space of harmonic functions in the unit ball that are
bounded by some increasing radial function $g(r)$ with $\lim_{r\rightarrow 1}
g(r)=+\infty$; these spaces are called growth spaces. We describe functions in
growth spaces by the Ces\`aro means of their expansions in harmonic polynomials
and apply this characterization to study coefficient multipliers between growth
spaces. A series of examples of multipliers is given and some results by A. L.
Shields and D. L. Williams and G. Bennett, D. A. Stegenga and R. M. Timoney are
generalized to harmonic functions in higher dimensions.
