math lessons - Integral transform

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.

Table of Integral Transforms
Transform	Symbol	Kernel	t₁	t₂
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 ).

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