Integral transform

From Academic Kids

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

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

Table of Integral Transforms
TransformSymbolKernelt1t2
Fourier transform

<math>\mathcal{F}<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>
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>
Hankel transform

<math>t\,J_\nu(ut)<math>

<math>0\,<math><math>\infty\,<math>
Abel transform

<math>\frac{t}{\sqrt{t^2-u^2}}<math>

<math>u\,<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>
Identity transform  

<math>\delta (u-t)\,<math>

<math>t_1<math>t_2>u\,<math>

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

  • A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.pt:Transformada integral

th:การแปลงเชิงปริพันธ์ zh:积分变换

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