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Difference between revisions of "Convolution transform"

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An integral transform of the type
 
An integral transform of the type
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$$
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F(x) = \int_{-\infty}^\infty G(x-t) f(t) dt \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026440/c0264401.png" /></td> </tr></table>
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026440/c0264402.png" /> is called the kernel of the convolution transform. For specific types of kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026440/c0264403.png" />, after suitable changes of variables, the convolution transform becomes the one-sided [[Laplace transform|Laplace transform]], the [[Stieltjes transform|Stieltjes transform]] or the [[Meijer transform|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]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Hirschman,  D.V. Widder,  "The convolution transform" , Princeton Univ. Press  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Y.A. Brychkov,  A.P. Prudnikov,  "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Hirschman,  D.V. Widder,  "The convolution transform" , Princeton Univ. Press  (1955)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  Y.A. Brychkov,  A.P. Prudnikov,  "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR>
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</table>
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Revision as of 17:00, 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. 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=18402
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