Transforms on time scales

Time scales theory provides a means for unifying and extending real and discrete analysis. Transforms play a crucial role in analysis in part because of transform methods for solving differential equations. Two of the most commonly used transforms are the Laplace and Fourier transforms. We define the Laplace transform for time scales noting that it is an extension of the Laplace transform for real numbers as well as a discrete transform. We give properties of the Laplace transform and discuss instances when results may not be generalized from the real case to times scales. Dynamic equations are solved in examples using the Laplace transform. Next we define the Fourier transform for time scales and discuss how it unifies the different types of Fourier analysis. Finally, we give a discussion on the possibilities for a general transform theory based on time scales analysis.