A mathematical background is very important and useful for all physical and engineering sciences. The connection between applied and mathematical research often leads to progress in both directions, due to natural synergy effects. The Acoustic Research Institute considers the investigation of the mathematical background of its numerous research projects, most prominently the signal processing aspects, as an important part of acoustic research.

*Application-oriented mathematics* develops theoretical results, motivated by application, in contrast to “applied mathematics” focusing on tools for the applied sciences. The application-oriented approach provides results significant both for the applied sciences and theoretical mathematics. The importance of application-oriented mathematics was acknowledged by the Viennese Technology and Science Fund arranging a specific research programme titled ‘Mathematics and …' and is a current research focus both of the Academy of Sciences and the city of Vienna.

Complex experimental designs generate empirical data and often lead to heuristic models with a modest mathematical basis. Mathematically precise statements considerably enhance the precision and stability of established algorithms and can already be implemented at an early stage of model generation. Therefore, mathematics supports the software development in the modelling stage as well as the implementation stage (stability, precision)

The Acoustics Research Institute has strengthened its research in this area in recent years, and will continue to do so. The following goals are set:

- Fundamental research in mathematical theory
- Application in psychoacoustical, phonetical and acoustical models.
- Development of efficient algorithms
- Particular focus on international, national and internal cooperation.

The cooperation of the group ’Mathematics and Signal Processing’ with the other groups of the Institute has been proven to be very fruitful for all partners and will be further strengthened. While the other groups get methods to solve their relevant problems, well-based in theory, the mathematicians can solve questions relevant for applications but still interesting in theory. This dialog increases the understanding of other fields enormously. It has allowed the successful application for the START-project 'FLAME: Frames and Linear Operators for Acoustical Modeling and Parameter Estimation' in 2011.

**Youtube Talks:**

- Nicki Holighaus - Time-Frequency Frames and Applications to Audio Analysis - Part 1
- Peter Balazs - February Fourier Talks 2014
- Peter Balazs - TUForMath Vortrag: Mathematik und Akustik
- Hans G. Feichtinger - Mathematical and Numerical Aspects of Frame Theory - Part 1 (showing the institute own software STx!)
- Georg Tauböck - WWTF Project INSIGHT

**Links:**

- Dictionary Learning for Sparse Audio Inpainting
- On Dual Frames