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 [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)