Reihe F Computational Fluid Dynamics and Data Analysis SHIFT GENERATED HAAR SPACES ON TRACK FIELDS

Abstract

Dedicated to the memory of Walter Hengartner 1 Abstract. The general aim is to show that G(z): = 1/z2 is never a universal Haar space generator for all compact sets K in C. For many cases that was already shown in papers by Hengartner & Opfer [5, 2002], [6, 2005]. The remaining cases are those for which K is convex (different from ellipses) and K = K ◦ , where K ◦ is the interior of K and K ◦ is the closure of K ◦ and where the boundary of K is smooth. We show for several cases of compact, convex sets that G is not a 2-dimensional Haar space generator for K implying that it is not a universal Haar space generator for K. We will be guided by a model of a track field: a rectangle with two half disks attached on two opposite sides of the rectangle. We also show, that the above G is not a 3-dimensional Haar space generator for all regular polygons (with smoothed vertices). The definition of Haar spaces and Haar space generators will be given in the main text. The paper contains as a byproduct a