ABSTRACT
The theory of convolution transformations was pioneered by I. Hirschman and D. Widder in the late https://www.w3.org/1998/Math/MathML"> 1940 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138752859/46b98bf4-f43d-4c1b-bad9-d831e2ef6414/content/eq5311.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in a series of papers https://www.w3.org/1998/Math/MathML"> [ 104 - 108 ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138752859/46b98bf4-f43d-4c1b-bad9-d831e2ef6414/content/eq5312.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> which were later compiled in a book form in the mid-1950s [103]. They showed, among other things, that for a certain class of kernels, the convolution transform encompasses a variety of other known integral transforms, such as the Laplace and the Stieltjes transforms. Chief among their discoveries of various properties of the convolution transform is the inversion formula.
