From an abstract point of view, the characterization of trigonometric polynomials amongst other functions F, in the harmonic analysis of the circle, is that for functions F in any of the typical function spaces, F is a trigonometric polynomial if and only if its Fourier coefficients
vanish for |n| large enough, and that this in turn is equivalent to the statement that all the translates
- F(t + θ)
by a fixed angle θ lie in a finite-dimensional subspace. One implication here is trivial, and the other, starting from a finite-dimensional invariant subspace, follows from complete reducibility of representations of T.
From this formulation, the general definition can be seen: for a representation ρ of K on a vector space V, a K-finite vector v in V is one for which the
for k in K span a finite-dimensional subspace. The union of all finite-dimension K-invariant subspaces is itself a subspace, and K-invariant, and consists of all the K-finite vectors. When all v are K-finite, the representation ρ itself is called K-finite.
Lectures on Lie Groups and Lie Algebras by Roger Carter, Graeme Segal and Ian Macdonald