Integral transform

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

$(Tf)(u)=\int _{{t_{1}}}^{{t_{2}}}f(t)\,K(t,u)\,dt.$

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.

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Table of Integral Transforms
TransformSymbolKernelt1t2
Fourier transform

${\mathcal {F}}$

${\frac {e^{{iut}}}{{\sqrt {2\pi }}}}$

$-\infty \,$$\infty \,$
Mellin transform

${\mathcal {M}}$

$t^{{u-1}}\,$

$0\,$$\infty \,$
Two-sided Laplace transform

${\mathcal {B}}$

$e^{{-ut}}\,$

$-\infty \,$$\infty \,$
Laplace transform

${\mathcal {L}}$

$e^{{-ut}}\,$

$0\,$$\infty \,$
Hankel transform

$t\,J_{\nu }(ut)$

$0\,$$\infty \,$
Abel transform

${\frac {t}{{\sqrt {t^{2}-u^{2}}}}}$

$u\,$$\infty \,$
Hilbert transform

${\mathcal {H}}$

${\frac {1}{\pi }}{\frac {1}{u-t}}$

$-\infty \,$$\infty \,$
Identity transform

$\delta (u-t)\,$

$t_{1}$t_{2}>u\,$

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).