We determine the class of Hilbert series H so that if M is a finitely
generated zero-dimensional R-graded module with the strong Lefschetz property,
then the tensor product of M and k[y]/(y^m) has the strong Lefschetz property
for y an indeterminate and all positive integers m if and only if the Hilbert
series of M is in H. Given two finite graded R-modules M and N with the strong
Lefschetz property, we determine sufficient conditions in order that the tensor
product of M and N has the strong Lefschetz property.