What is "integral transform"

n. (context mathematics English) Any transform of the form:

In mathematics, an integral transform is any transform T of the following form:

$$(Tf)(u) = \int \limits_{t_1}^{t_2} K(t, u)\, f(t)\, dt$$

The input of this transform is a function f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator.

There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform.

Some kernels have an associated inverse kernel K(u, t) which (roughly speaking) yields an inverse transform:

$$f(t) = \int \limits_{u_1}^{u_2} K^{-1}( u,t )\, (Tf)(u)\, du$$

A symmetric kernel is one that is unchanged when the two variables are permuted .