Corresponding Banach spaces on time scales

摘要

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.