In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. From the point of view of functional analysis it therefore has a well-understood abstract eigenvalue theory. In this case that is supported by a computational theory, including the Fredholm determinants. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Erik Ivar Fredholm (April 7, 1866 - August 17, 1927) was a Swedish mathematician who established the modern theory of integral equations. ... In mathematics, a Fredholm operator is a bounded linear operator between two Banach spaces whose range is closed and whose kernel and cokernel are finite-dimensional. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... A complex analytic function which generalizes the characteristic polynomial of a matrix. ...

An inhomogeneous Fredholm equation of the first kind is written: Homogeneous is an adjective that has several meanings. ...

and the problem is, given the continuous kernel function K(t,s), and the function g(t), to find the function f(s). In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ...

If the kernel has the specific form K(t-s) and the limits of integration are , the righr hand side of the equaiton can be rewritten as a convolution of the functions K and f and therefore the solution will be given by

where and are the direct and inverse Fourier transforms respectively.

An inhomogeneous Fredholm equation of the second kind is essentially a form of the eigenvalue problem for the above equation: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

and the problem is again, given the kernel K(t,s), and the function g(t), find the function f(s). The kernel K is a compact operator (to show this one relies on equicontinuity). It therefore has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0. This underlies the theory of the equation. In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a... In mathematical analysis, a sequence of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood (a precise definition appears below). ... In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

See also

• Fredholm kernel
• Liouville-Neumann series

In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. ... In mathematics, the Liouville-Neumann series is an infinite series defined as which is a unique, continuous solution of a Fredholm integral equation of the second kind. ...

References

• Integral Equations at EqWorld: The World of Mathematical Equations.
• A.D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4