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Concept
Fredholm theory
Résumé
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. The theory is named in honour of Erik Ivar Fredholm.
The following sections provide a casual sketch of the place of Fredholm theory in the broader context of operator theory and functional analysis. The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details.
Much of Fredholm theory concerns itself with the following integral equation for f when g and K are given:
This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is asked to solve the differential equation
where the function f is given and g is unknown. Here, L stands for a linear differential operator.
For example, one might take L to be an elliptic operator, such as
in which case the equation to be solved becomes the Poisson equation.
A general method of solving such equations is by means of Green's functions, namely, rather than a direct attack, one first finds the function such that for a given pair x,y,
where δ(x) is the Dirac delta function.
The desired solution to the above differential equation is then written as an integral in the form of a Fredholm integral equation,
The function K(x,y) is variously known as a Green's function, or the kernel of an integral. It is sometimes called the nucleus of the integral, whence the term nuclear operator arises.
In the general theory, x and y may be points on any manifold; the real number line or m-dimensional Euclidean space in the simplest cases. The general theory also often requires that the functions belong to some given function space: often, the space of square-integrable functions is studied, and Sobolev spaces appear often.
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