K-finite
Encyclopedia
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 space
s, 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a K-finite function is a type of generalized trigonometric polynomial
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...
. Here K is some compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
, 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
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
of the circle, is that for functions F in any of the typical function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
s, 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
Invariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
, 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.