After successful completion of the course, students are able to appreciate and rate the practical value of methods for inversion of various integral transforms. The course also aims at training the ability to communicate essentials of the theory to engineers with a more moderate mathematical background. Besides, the participants will learn about several results from ‘pure‘ mathematics which turned out to be of eminent practical use decades later. An English version of the material can be provided on demand (e.g. for Erasmus students).
Several invertible integral transforms are presented, in particular the classical two-dimensional transform and its inversion formula introduced by J. Radon in 1917 which was used for the development of parallel beam tomography in the early 1970’s (A. Hounsfield and A. Cormack were awarded the Nobel prize for their pioneering work in 1979). In recent years the Radon transform has been used for automatized detection of certain irregularities in surface structures. Many generalizations are known today. A class of inversion formulas is based on Riesz potentials. In computerized tomography, since the 1990’s interest has shifted to cone beam projection methods. A major theoretical breakthrough in this field was achieved in 2002 by A. Katsevich who established a three-dimensional integral transform together with an exact inversion formula and a stable algorithm of backprojection type which makes use of the fast Fourier transform. We also discuss the famous identity in functional analysis found by Calderón in 1960 which was rediscovered by theoretical physicists two decades later and has proved as fundamental in wavelet theory. The tools of the proofs comprise a tour through analysis with Fourier transform in prominent place.
Handouts are provided which will sum up to a textbook. The results are illustrated by numerous Maple graphics. Furthermore, Maple programs help understanding more intricate derivations. Basic ideas and structures of proofs are routinely emphasized and commented.
Oral examination
Not necessary
Bachelor level (Mathematische Grundausbildung / 1. Studienabschnitt)
