Integral transform

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

<math> (Tf)(u) = \int_{t_1}^{t_2} K(t, u)\, f(t)\, dt.</math>

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 <math>K^{-1}(u,t)</math> which (roughly speaking) yields an inverse transform:

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

Contents

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Table of transforms

Table of integral transforms
Transform Symbol <math>K</math> t1 t2 <math>K^{-1}</math> u1 u2
Fourier transform <math>\mathcal{F}</math> <math>\frac{e^{-iut}}{\sqrt{2 \pi}}</math> <math>-\infty\,</math> <math>\infty\,</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> <math>\frac{t^{-u}}{2\pi i}\,</math> <math>c\!-\!i\infty</math> <math>c\!+\!i\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> <math>\frac{e^{+ut}}{2\pi i}</math> <math>c\!-\!i\infty</math> <math>c\!+\!i\infty</math>
Hankel transform <math>t\,J_\nu(ut)</math> <math>0\,</math> <math>\infty\,</math> <math>u\,J_\nu(ut)</math> <math>0\,</math> <math>\infty\,</math>
Abel transform <math>\frac{2t}{\sqrt{t^2-u^2}}</math> <math>u\,</math> <math>\infty\,</math> <math>\frac{-1}{\pi\sqrt{u^2\!-\!t^2}}\frac{d}{du}</math> <math>t\,</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> <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<u\,</math> <math>t_2>u\,</math> <math>\delta (t-u)\,</math> <math>u_1\!<\!t</math> <math>u_2\!>\!t</math>
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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 equations 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.

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See also

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References

gl:Transformada integral nl:Integraaltransformatie pt:Transformada integral th:การแปลงเชิงปริพันธ์ zh:积分变换

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