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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
vanish for |n| large enough, and that this in turn is equivalent to the statement that all the translates
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.