Fredholm Theory - Homogeneous Equations

Homogeneous Equations

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 is the Dirac delta function. The desired solution to the differential equation is then written as

This integral is written in the form of a Fredholm integral equation. The function 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.

The actual function space used is often determined by the solutions of the eigenvalue problem of the differential operator; that is, by the solutions to

where the are the eigenvalues, and the are the eigenvectors. The set of eigenvectors span a Banach space, and, when there is a natural inner product, then the eigenvectors span a Hilbert space, at which point the Riesz representation theorem is applied. Examples of such spaces are the orthogonal polynomials that occur as the solutions to a class of second-order ordinary differential equations.

Given a Hilbert space as above, the kernel may be written in the form

where is the dual to . In this form, the object is often called the Fredholm operator or the Fredholm kernel. That this is the same kernel as before follows from the completeness of the basis of the Hilbert space, namely, that one has

Since the are generally increasing, the resulting eigenvalues of the operator are thus seen to be decreasing towards zero.