ABSTRACT
In this chapter, we give an expository review of some integral transforms, Parseval–Goldstein-type relationships and their applications to integral transforms, and some well-known differential equations. The generalized Stieltjes transform can be obtained by iterating the Laplace transform. This observation leads to the Parseval-type theorem relating the Laplace transform to the generalized Stieltjes transform. We present identities and Parseval-type relationships resulting from the results for some integral transforms and fractional integrals. In an analogous manner, it is shown that the Widder transform can be obtained as an iteration of the L2-transform. This observation produces a Parseval-type relationship relating the L2-transform to the Widder transform and identities for these transforms. As application of the L2-transform, we solve the well-known Bessel’s differential equation and Hermite’s differential equation using the transform.
