Fredholm Integral Equation of the First Kind

Equation of the first kind

Integral equations, most generally, are common and take many specific forms (Fourier, Laplace, Hankel, etc.). They each differ in their kernels (defined below). What is distinctive about Fredholm integral equations is that they are integral equations in which the integration limits are constants (they do not include the variable). This is contrast to Volterra integral equations.

A homogeneous Fredholm equation of the first kind is written as:

General theory

The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel K is a compact operator, known as the Fredholm operator. Compactness may be shown by invoking equicontinuity. As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0.

[edit]Applications

Fredholm equations arise naturally in the theory of signal processing, most notably as the famous spectral concentration problem popularized by David Slepian. They also commonly arise in linear forward modeling and inverse problems.