New Concepts in Frame Theory Motivated by Acoustical Applications

Preface Application-oriented mathematics develops theoretical results and new mathematical concepts, motivated by application, in contrast to " applied mathematics " focusing just on providing and applying mathematical tools for the applied sciences. The application-oriented approach produces results significant both for mathematics and the applied sciences. In this context we developed new concepts in frame theory motivated by signal processing and acoustical applications. Frames are generalizations of bases, and give more freedom for the analysis and modification of information. The concept of frames is a theoretical background for signal processing. On the other hand, signal processing algorithms and processes are essential for application in audio and acoustics. Linking the mathematical frame theory, the signal processing algorithms, their implementations and finally acoustical applications leads to a very promising, synergetic combination of research in different fields, which has not been fully exploited yet. To establish that link a thorough investigation of the theory is important. So we have investigated topics in frame theory, extending the standard mathematical concepts. As a particular case of analysis and synthesis systems we have researched mathematical topics in time-frequency analysis. Furthermore a big focus was the mathematical theory of multipliers, which are operators created by combining frame analysis, multiplication and re-synthesis. To show that frame theory is important for applications we have included two applied topics, which both apply Gabor frame multipliers in acoustical projects. The focus of our work is also the focus of this habilitation thesis and can be summarized by the following grouping: