# Convolution transform

2010 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://www.encyclopediaofmath.org/index.php?title=Convolution_transform&oldid=37030
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