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Difference between revisions of "Convolution transform"

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An integral transform of the type
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An [[integral transform]] of the type
 
$$
 
$$
 
F(x) = \int_{-\infty}^\infty G(x-t) f(t) dt \ .
 
F(x) = \int_{-\infty}^\infty G(x-t) f(t) dt \ .
 
$$
 
$$
  
The function $G$ is called the kernel of the convolution transform. For specific types of kernels $G$, after suitable changes of variables, the convolution transform becomes the one-sided [[Laplace transform]], the [[Stieltjes transform]] or the [[Meijer transform]]. The inversion of a convolution transform is realized by linear differential operators of infinite order that are invariant with respect to a shift.
+
The function $G$ is called the kernel of the convolution transform (cf. [[Kernel of an integral operator]]). For specific types of kernels $G$, after suitable changes of variables, the convolution transform becomes the one-sided [[Laplace transform]], the [[Stieltjes transform]] or the [[Meijer transform]]. The inversion of a convolution transform is realized by linear differential operators of infinite order that are invariant with respect to a shift.
  
 
The convolution transform is also defined for certain classes of generalized functions (see [[#References|[2]]]).
 
The convolution transform is also defined for certain classes of generalized functions (see [[#References|[2]]]).

Revision as of 17:01, 20 December 2015

An integral transform of the type $$ F(x) = \int_{-\infty}^\infty G(x-t) f(t) dt \ . $$

The function $G$ is called the kernel of the convolution transform (cf. Kernel of an integral operator). For specific types of kernels $G$, after suitable changes of variables, the convolution transform becomes the one-sided Laplace transform, the Stieltjes transform or the Meijer transform. The inversion of a convolution transform is realized by linear differential operators of infinite order that are invariant with respect to a shift.

The convolution transform is also defined for certain classes of generalized functions (see [2]).

References

[1] I.I. Hirschman, D.V. Widder, "The convolution transform" , Princeton Univ. Press (1955)
[2] Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)
How to Cite This Entry:
Convolution transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convolution_transform&oldid=37028
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article