Difference between revisions of "Convolution transform"
From Encyclopedia of Mathematics
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| − | An integral transform of the type | + | {{MSC|44A35}} |
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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]]]). | ||
Latest revision as of 17:03, 20 December 2015
2020 Mathematics Subject Classification: Primary: 44A35 [MSN][ZBL]
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
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