"Because the links between a convolution integral and a Laplace or are so important... we briefly present Borel's (1899) work on a "Laplace like" transform. Note Mellin's work (1896)... was unknown to Borel... Borel defined two functions f(z) and g(z) by their following Laplace integrals...:f(z) = \int\limits_{0}^{+\infty}F(u)e^{-u/z}\, \frac{du}{z}; \quad g(z) = \int\limits_{0}^{+\infty}G(v)e^{-v/z}\, \frac{dv}{z}and then showed that the convolution integral is H(x) = \int\limits_{0}^{x}F(t)G(x-t)\,dt. The Laplace transform of the convolution integral H(x) reduced to a simple product of the two separate transforms f(z) and g(z). Borel failed to see all the possibilities of his theorem. Volterra... also did not see the possible uses... But... in 1920, Doetsch produced a doctoral thesis... on Borel's summability theory of diverging series. Doetsch knew Borel's proof and was able to introduce modern, proper mathematical ideas on convolution integrals and Laplace transforms. The word Faltung was first introduced by Doetsch and Bernstein in 1920. The Laplace transform... and the Fourier transform... are both adequate tools for evaluating a convolution integral. ...Doetsch would introduce the convolution integral by analogy with a between two ..."