| dc.contributor.advisor | Bahns, Dorothea Prof. Dr. | |
| dc.contributor.author | Blobel, Burkhard | |
| dc.date.accessioned | 2021-03-29T09:36:45Z | |
| dc.date.available | 2021-04-04T00:50:07Z | |
| dc.date.issued | 2021-03-29 | |
| dc.identifier.uri | http://hdl.handle.net/21.11130/00-1735-0000-0008-57D8-3 | |
| dc.identifier.uri | http://dx.doi.org/10.53846/goediss-8523 | |
| dc.identifier.uri | http://dx.doi.org/10.53846/goediss-8523 | |
| dc.language.iso | eng | de |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | |
| dc.subject.ddc | 510 | de |
| dc.title | Continuous Wavelet Transformation on Homogeneous Spaces | de |
| dc.type | doctoralThesis | de |
| dc.contributor.referee | Bahns, Dorothea Prof. Dr. | |
| dc.date.examination | 2020-12-11 | |
| dc.description.abstracteng | The classical Continuous Wavelet Transformation (cCWT) is an important and well-studied tool in signal processing and data analysis. Because of its deep connection to representation theory, it has come into focus of pure mathematics. It has been shown that it constitutes a unitary equivalence between two (necessarily equivalent) representations of some Lie group. One of those representations is defined on a homogeneous space of the Lie group, or, in other words, it is an induced representation. The other one is a subrepresentation of the left regular representation of the Lie group. This result inspired theorists to generalizations of the cCWT, which we call Continuous Wavelet Transformations (CWT). Further analysis showed that representations defined on homogeneous spaces of some Lie group admit a CWT only under certain conditions. The first one is that the representation has to be (quasi-)equivalent to a subrepresentation of the left regular representation. For nonunimodular groups this condition is already sufficient. If, however, the group is unimodular then another one is needed, which states that the representation has to be "small" enough in order to avoid the appearance of infinities. Since those conditions do not depend on the differential structure of the group, these results apply not only to Lie groups but also to second countable locally compact groups.
The focus of this thesis is on unitary (not necessarily irreducible) representations of second countable locally compact groups defined on homogeneous spaces which do not admit a CWT. The aim is to alter the definition of a CWT in such a way that those representations admit a generalized Continuous Wavelet Transformation (gCWT). In the first main result of this thesis it is shown that any representation can be decomposed into a subrepresentation which is quasi-equivalent to a subrepresentation of the left regular representation and a second one which is disjoint to the left regular representation. The importance of this result lies in the fact that both parts can, and actually need to, be treated separately. In the second main theorem it is proved that any representation being quasi-equivalent to a subrepresentation of the left regular representation which is too "large" can be decomposed into smaller parts which admit CWTs. In that way it is possible to bypass the appearance of the infinities. As mentioned above, this procedure is only necessary for unimodular groups. On the other hand, if the representation is disjoint to the left regular representation then the situation is considerably more challenging. Nevertheless, in the third main theorem of this thesis it is shown that for a certain class of groups these representations can be treated by adapting an approach which is also known as coherent state transformation. The groups considered here are extensions of some Lie group which acts on a real finite-dimensional vector space by this vector space. The representations which are considered in that section are induced representations, i.e., defined on homogeneous spaces of the group.
In conclusion, by proposing new approaches for gCWT, the scope of application of Continuous Wavelet Transformations has been extended and it offers possibilities for new developments in applied and pure mathematics. | de |
| dc.contributor.coReferee | Schick, Thomas Prof. Dr. | |
| dc.subject.eng | Continuous Wavelet Transformation | de |
| dc.subject.eng | Coherent State Transformation | de |
| dc.subject.eng | Abstract Harmonic Analysis | de |
| dc.subject.eng | Representation Theory of Locally Compact Groups | de |
| dc.identifier.urn | urn:nbn:de:gbv:7-21.11130/00-1735-0000-0008-57D8-3-8 | |
| dc.affiliation.institute | Fakultät für Mathematik und Informatik | de |
| dc.subject.gokfull | Mathematics (PPN61756535X) | de |
| dc.description.embargoed | 2021-04-04 | |
| dc.identifier.ppn | 175271749X | |