Integral transform

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.

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

Table of Integral Transforms
Transform	Symbol	Kernel	t₁	t₂
Fourier transform	<math>\mathcal{F}<math>	<math>\frac{e^{iut}}{\sqrt{2 \pi}}<math>	<math>-\infty\,<math>	<math>\infty\,<math>
Mellin transform	<math>\mathcal{M}<math>	<math>t^{u-1}\,<math>	<math>0\,<math>	<math>\infty\,<math>
Two-sided Laplace transform	<math>\mathcal{B}<math>	<math>e^{-ut}\,<math>	<math>-\infty\,<math>	<math>\infty\,<math>
Laplace transform	<math>\mathcal{L}<math>	<math>e^{-ut}\,<math>	<math>0\,<math>	<math>\infty\,<math>
Hankel transform		<math>t\,J_\nu(ut)<math>	<math>0\,<math>	<math>\infty\,<math>
Abel transform		<math>\frac{t}{\sqrt{t^2-u^2}}<math>	<math>u\,<math>	<math>\infty\,<math>
Hilbert transform	<math>\mathcal{H}<math>	<math>\frac{1}{\pi}\frac{1}{u-t}<math>	<math>-\infty\,<math>	<math>\infty\,<math>
Identity transform		<math>\delta (u-t)\,<math>	<math>t_1	<math>t_2>u\,<math>

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