Fredholm theory

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 finding solutions for the integral equation

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 instead attempts to solve the equation

where δ(x) is the Dirac delta function.

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