A set of basis functions are complete if you can manufacture all functions on the group with them.

There are associated orthogonality conditions:

Sum(-infinity,infinity,Xm*(g)Xm(g'))=delta(g,g')

f(g)=Sum(m,cmXm(g)) where cm=Integral(Xm*(g)f(g)dg)

source

jl@crush.caltech.edu index