Editing Integral transform (section)

==General form==
An integral transform is any [[Transformation (function)|transform]] ''<math>T</math>'' of the following form:

:<math>(Tf)(u) = \int_{t_1}^{t_2} f(t)\, K(t, u)\, dt</math>

The input of this transform is a [[function (mathematics)|function]] ''<math>f</math>'', and the output is another function ''<math>Tf</math>''. An integral transform is a particular kind of mathematical [[Operator (mathematics)|operator]].

{{anchor|kernel|kernel function|integral kernel}}There are numerous useful integral transforms. Each is specified by a choice of the function <math>K</math> of two [[Variable (mathematics)|variables]], that is called the '''kernel''' or '''nucleus''' of the transform.

Some kernels have an associated ''inverse kernel'' <math>K^{-1}( u,t )</math> which (roughly speaking) yields an inverse transform:

:<math>f(t) = \int_{u_1}^{u_2} (Tf)(u)\, K^{-1}( u,t )\, du</math>

A ''symmetric kernel'' is one that is unchanged when the two variables are permuted; it is a kernel function ''<math>K</math>'' such that <math>K(t, u) = K(u, t)</math>. In the theory of integral equations, symmetric kernels correspond to [[self-adjoint operators]].<ref> Chapter 8.2, Methods of Theoretical Physics Vol. I (Morse & Feshbach)</ref>