The Laplace transform on time scales revisited

In this work, we reexamine the time scale Laplace transform as defined by Bohner and Peterson [M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001; M. Bohner, A. Peterson, Laplace transform and Z-transform: Unification and extension, Methods Appl. Anal. 9 (1) (2002) 155–162]. In particular, we give conditions on the class of functions which have a transform, develop an inversion formula for the transform, and further, we provide a convolution for the transform. The notion of convolution leads to considering its algebraic structure—in particular the existence of an identity element—motivating the development of the Dirac delta functional on time scales. Applications and examples of these concepts are given.