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Concept# Eigenfunction

Summary

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as
for some scalar eigenvalue The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions.
An eigenfunction is a type of eigenvector.
In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function f is an eigenfunction of D if it satisfies the equation
where λ is a scalar. The solutions to Equation may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ1, λ2, ... or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both.
Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or geometric multiplicity.
A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space C∞ of infinitely differentiable real or complex functions of a real or complex argument t. For example, consider the derivative operator with eigenvalue equation
This differential equation can be solved by multiplying both sides by and integrating. Its solution, the exponential function
is the eigenfunction of the derivative operator, where f0 is a parameter that depends on the boundary conditions.

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In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as for some scalar eigenvalue The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector.

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In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the

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