Abstract and Applied Analysis
Volume 2007 (2007), Article ID 58373, 24 pages
The Convolution on Time Scales
1Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla 65401, MO, USA
2Department of Mathematics, Atilim University, Incek 06836, Ankara, Turkey
Received 19 March 2007; Accepted 24 May 2007
Academic Editor: Allan C. Peterson
Copyright © 2007 Martin Bohner and Gusein Sh. Guseinov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The main theme in this paper is an initial value problem containing a dynamic
version of the transport equation. Via this problem, the delay (or shift) of a function defined on a
time scale is introduced, and the delay in turn is used to introduce the convolution of two functions
defined on the time scale. In this paper, we give some elementary properties of the delay and of the
convolution and we also prove the convolution theorem. Our investigation contains a study of the
initial value problem under consideration as well as some results about power series on time scales.
As an extensive example, we consider the -difference equations case.