# K-finite

In mathematics, a **K-finite function** is a type of generalized trigonometric polynomial. Here *K* is some compact group, and the generalization is from the circle group *T*.

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

*a*_{'n}

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

- ρ(
*k*).*v*

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.

## References

Lectures on Lie Groups and Lie Algebras by Roger Carter, Graeme Segal and Ian Macdonald