Difference between revisions of "Convolution transform"
From Encyclopedia of Mathematics
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An integral transform of the type | 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 function | ||
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 & Breach (1989) (Translated from Russian)</TD></TR></table> | + | <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 & Breach (1989) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
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
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