Corresponding Banach spaces on time scales

We will provide a short introduction to the calculus on a time scale T, in order to make the reader familiar with the basics. Then we intend to have a closer look at the so-called “cylinder transform” ξμ which maps a positively regressive function p:T→R to another function p˜:T→R. It will turn out that, under certain conditions, this cylinder transform acts as an isometry between two normed spaces. Therefore, we obtain a two-fold generalization of the well-known Banach and Hilbert spaces of functions in continuum analysis. Finally, we shall give some examples concerning this structure of corresponding spaces-for instance an example of orthogonal polynomials on equidistant lattices. In order to achieve this, we shall state a theorem on how to take orthogonality theory over from a Hilbert space to its corresponding Hilbert space.