List of integral transforms

Summary

Each integral transform has the form $(Tf)(u) = \int_{t_1}^{t_2} f(t)\, K(t, u)\, dt$ , and it is fixed by its kernel and its interval of integration. The table below lists common transforms with one standard normalisation; books differ in constants and signs.

Transform	Kernel $K(t, u)$	Interval
Laplace transform	$e^{-ut}$	$(0, \infty)$
Two-sided Laplace transform	$e^{-ut}$	$(-\infty, \infty)$
Fourier transform	$e^{-2\pi i u t}$	$(-\infty, \infty)$
Mellin transform	$t^{u - 1}$	$(0, \infty)$
Hankel transform	$t\, J_\nu(ut)$	$(0, \infty)$
Stieltjes transform	$1/(u + t)$	$(0, \infty)$
Hilbert transform	$\frac{1}{\pi}\frac{1}{u - t}$ (principal value)	$(-\infty, \infty)$
Weierstrass transform	$\frac{1}{\sqrt{4\pi}} e^{-(u - t)^2/4}$	$(-\infty, \infty)$
Hartley transform	$\frac{1}{\sqrt{2\pi}}(\cos ut + \sin ut)$	$(-\infty, \infty)$

Here $J_\nu$ is the Bessel function of the first kind of order $\nu$ . Some transforms do not fit this one-line form. The Radon transform integrates over lines, and the Abel transform has a variable domain of integration.

Tag	`2P4RDS0A6X`
URI	`m2://2P4RDS0A6X`
Kind	concept
Writing	W0 stub
MSC	44A05, 44A15, 44A20

Tag

2P4RDS0A6X

URI

m2://2P4RDS0A6X

Kind

concept

Writing

W0 stub

MSC

44A05, 44A15, 44A20

List of integral transforms

Summary

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