Editing Fredholm integral equation (section)

==Equation of the first kind==

A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the [[Volterra integral equation]] which has variable integral limits.

An [[inhomogeneous equation|inhomogeneous]] Fredholm equation of the first kind is written as

{{Equation box 1
|indent =::
|equation = <math>g(t)=\int_a^b K(t,s)f(s)\,\mathrm{d}s   ~,</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}

and the problem is, given the continuous [[kernel (integral equation)|kernel]] function <math>K</math> and the function <math>g</math>, to find the function <math>f</math>.

An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely <math> K(t,s)=K(t-s)</math>, and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a [[convolution]] of the functions <math>K</math> and <math>f</math> and therefore, formally, the solution  is given by
:<math>f(s) =  \mathcal{F}_\omega^{-1}\left[
{\mathcal{F}_t[g(t)](\omega)\over
\mathcal{F}_t[K(t)](\omega)}
\right]=\int_{-\infty}^\infty {\mathcal{F}_t[g(t)](\omega)\over
\mathcal{F}_t[K(t)](\omega)}e^{2\pi i \omega s} \mathrm{d}\omega </math>
where <math>\mathcal{F}_t</math> and <math>\mathcal{F}_\omega^{-1}</math> are the direct and inverse [[Fourier transforms]], respectively. This case would not be typically included under the umbrella of Fredholm integral equations, a name that is usually reserved for when the integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of convolution with <math>K</math> contains the range of <math>\mathcal{F}{K}</math>, which is usually a non-countable set, whereas compact operators have discrete countable spectra).