Integral transform

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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}<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).

See also

External links

Bibliography

pt:Transformada integral th:การแปลงเชิงปริพันธ์ zh:积分变换