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.

Table of Integral Transforms
Transform	Symbol	Kernel	t₁	t₂
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		$\delta (u-t)\,$	$t_1<u\,$	$t_2>u\,$

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).

External links

Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.

Bibliography

A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.

Integral transform

See also

External links

Bibliography

See also: Integral transform, Abel transform, Dirac delta function, Fourier transform, Function (mathematics), Generalized function, Hankel transform, Hilbert transform, Kernel (integral operator), Laplace transform