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.
| Transform | Symbol | Kernel | t1 | t2 |
|---|---|---|---|---|
| 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
- Tables of Integral Transforms (http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm) at EqWorld: The World of Mathematical Equations.
Bibliography
- A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.pt:Transformada integral
