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, Birkhauser, Boston, 2001; M. Bohner, A. Peterson, Laplace transform and Z-transforrn: 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. (C) 2006 Elsevier Inc. All rights reserved.