View source for Convolution transform

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 [[#References|[2]]]).

====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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