# Convolution transform

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=37030