PlanetMath: integral transform

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integral transform

(Definition)

A generic integral transform takes the form

$\displaystyle F(p) = \int_\alpha^\beta K(p, t) f(t)dt,$

with

being the transform parameter.

Note that the transform takes a function and produces a new function .

The function is called the kernel of the transform. The kernel of an integral transform, along with the limits $\alpha$ and $\beta$ , distinguish a particular integral transform from another.

Examples

Laplace transform

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

$\displaystyle F(p) = \int\limits_0^\infty e^{-pt} f(t) dt.$
Laplace-Carson transform

$\displaystyle \alpha = 0,\; \beta = \infty,\; K(p,t) = pe^{-pt},$

$\displaystyle F(p) = \int\limits_0^\infty pe^{-pt} f(t) dt.$
Fourier transform

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

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

"integral transform" is owned by PrimeFan. [ full author list (4) | owner history (5) ]

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See Also: contour integral, group homomorphism

Also defines:

kernel

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Cross-references: Fourier transform, Laplace transform, function, parameter
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This is version 6 of integral transform, born on 2002-04-05, modified 2008-06-19.
Object id is 2815, canonical name is IntegralTransform.
Accessed 7076 times total.

Classification:

AMS MSC:

65R10 (Numerical analysis :: Integral equations, integral transforms :: Integral transforms)

Pending Errata and Addenda

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