Integral Transform

(Tf)(u) = \int_{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''.

There are several useful integral transforms.
Each transform corresponds to a different choice of the function ''K'', which is called the kernel or the '''nucleus''' of the transform. With each kernel is associated an inverse kernel

K^{-1}(u,t)

which (roughly speaking) yields an inverse transform:

:

f(t) = \int_{u_1}^{u_2} K^{-1}(u,t)\, (Tf)(u)\, du.

TABLE OF TRANSFORMS

Table of integral transforms
Transform	Symbol	$K$	t₁	t₂	$K^{-1}$	u₁	u₂
Fourier Transform	$\mathcal{F}$	$rac{e^{-iut}}{\sqrt{2 \pi}}$	$-\infty\,$	$\infty\,$	$rac{e^{+iut}}{\sqrt{2 \pi}}$	$-\infty\,$	$\infty\,$
Mellin Transform	$\mathcal{M}$	$t^{u-1}\,$	$0\,$	$\infty\,$	$rac{t^{-u}}{2\pi i}\,$	$c\!-\!i\infty$	$c\!+\!i\infty$
Two-sided Laplace transform	$\mathcal{B}$	$e^{-ut}\,$	$-\infty\,$	$\infty\,$
Laplace Transform	$\mathcal{L}$	$e^{-ut}\,$	$0\,$	$\infty\,$	$rac{e^{+ut}}{2\pi i}$	$c\!-\!i\infty$	$c\!+\!i\infty$
Hankel Transform		$t\,J_ u(ut)$	$0\,$	$\infty\,$	$u\,J_ u(ut)$	$0\,$	$\infty\,$
Abel Transform		$rac{2t}{\sqrt{t^2-u^2}}$	$u\,$	$\infty\,$	$rac{-1}{\pi\sqrt{u^2\!-\!t^2}}rac{d}{du}$	$t\,$	$\infty\,$
Hilbert Transform	$\mathcal{H}$	$rac{1}{\pi}rac{1}{u-t}$	$-\infty\,$	$\infty\,$	$rac{1}{\pi}rac{1}{u-t}$	$-\infty\,$	$\infty\,$
Identity Transform		$\delta (u-t)\,$	$t_1$	$t_2>u\,$	$\delta (t-u)\,$	$u_1\!<\!t$	$u_2\!>\!t$

GENERAL THEORY

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 ).

The general theory of such Integral Equation s is known as Fredholm Theory . In this theory, the kernel is understood to be a Compact Operator acting on a Banach Space of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm Operator , the Nuclear Operator or the Fredholm Kernel .

SEE ALSO

List Of Transforms

List Of Fourier-related Transforms

Kernel Trick

Nachbin's Theorem

REFERENCES

A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4

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

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Integral Transform