integral transform

A generic integral transform takes the form

$$F(p)={\int }_{\alpha }^{\beta }K(p,t)f(t)𝑑t,$$

with $p$ being the transform parameter.

Note that the transform takes a function $f(t)$ and produces a new function $F(p)$.

The function $K(p,t)$ is called the kernel of the transform. The kernel of an integral transform, along with the limits (http://planetmath.org/DefiniteIntegral) $\alpha$ and $\beta$, distinguish a particular integral transform from another.

Examples

• Laplace transform

  $$\alpha =0,\beta =\mathrm{\infty },K(p,t)=e^{-pt},$$

  $$F(p)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-pt}f(t)𝑑t.$$

• Laplace-Carson transform

  $$\alpha =0,\beta =\mathrm{\infty },K(p,t)=p{e}^{-pt},$$

  $$F(p)=\underset{0}{\overset{\mathrm{\infty }}{\int }}p{e}^{-pt}f(t)𝑑t.$$

• Fourier transform

  $$\alpha =-\mathrm{\infty },\beta =\mathrm{\infty },K(p,t)=\frac{1}{\sqrt{2\pi }}{e}^{-ipt},$$

  $$F(p)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-ipt}f(t)𝑑t.$$

Title	integral transform
Canonical name	IntegralTransform
Date of creation	2013-03-22 12:34:03
Last modified on	2013-03-22 12:34:03
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	10
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 65R10
Related topic	ContourIntegral
Related topic	GroupHomomorphism
Defines	kernel
Defines	transform parameter