Integral transform

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

$(Tf)(u) = \int f(t)\, g(t, u)\, dt.$

Tf is a function of a parameter, denoted u in the equation above. Thus, an integral transform maps one function into another which is a function of the parameter.

There are several useful integral transforms. Each transform corresponds to a different choice of the function g, which is called the kernel of the transform.

Table of Integral Transforms
Transform	Symbol	Kernel
Laplace transform	$\mathcal{L}$	$e^{-ut}\,$
Fourier transform	$\mathcal{F}$	$\frac{e^{iut}}{\sqrt{2 \pi}}$
Hilbert transform	$\mathcal{H}$	$\frac{1}{\pi}\frac{1}{x-t}$
Mellin transform	$\varphi\,$	$t^{z-1}\,$
Identity transform		$\delta (u-t)\,$

Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem).

Integral transform

See also