In mathematics, a K-finite function is a type of generalized

trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...

. Here ''K'' is some

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural ge ...

, and the generalization is from the

circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ...

''T''. From an abstract point of view, the characterization of trigonometric polynomials amongst other functions ''F'', in the

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...

of the circle, is that for functions ''F'' in any of the typical

function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...

s, ''F'' is a trigonometric polynomial if and only if its

Fourier coefficient A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

s :''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 In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General descr ...

, follows from

complete reducibility In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''s ...

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

* {{citation , last1 = Carter , first1 = Roger W. , authorlink1 = Roger Carter (mathematician) , last2 = Segal , first2 = Graeme , author-link2 = Graeme Segal , last3 = Macdonald , first3 = Ian , author-link3 = Ian G. Macdonald , title = Lectures on Lie Groups and Lie Algebras , publisher = Cambridge University Press , isbn = 0-521-49579-2 Representation theory of groups