### Objective:

The applications involving signal processing algorithms (like adaptive or time variant filters) are numerous. If the STFT, the Short Time Fourier Transformation, is used in its sampled version, the Gabor transform, the use of Gabor multipliers creates a possibility to construct a time-variant filter. The Gabor transform is used to calculate time frequency coefficients, which are multiplied with a fixed time-frequency mask. Then the result is synthesized. If another way of calculating these coefficients is chosen or if another synthesis is used, many modifications can still be implemented as multipliers. For example, it seems quite natural to define wavelet multipliers. Therefore, for this case, it is quite natural to continue generalizing and look at multipliers with frames lacking any further structure.

### Method:

Therefore, for Bessel sequences, the investigation of operators

M = ∑ mk < f , ψk > φk

where the analysis coefficients, < f , ψk >, are multiplied by a fixed symbol mk before resynthesis (with φk), is very natural and useful. These are the Bessel multipliers investigated in this project. The goal of this project is to set the mathematical basis to unify the approach to the Bessel multipliers for all possible analysis / synthesis sequences that form a Bessel sequence.

### Application:

Bessel sequences and frames are used in many applications. They have the big advantage of allowing the possibility to interpret the analysis coefficients. This makes the formulation of a multiplier concept for other analysis / synthesis systems very profitable. One such system involves gamma tone filter banks, which are mainly used for analysis based on the auditory system.

### Publications:

• Balazs, P. (2007), "Basic Definition and Properties of Bessel Multipliers", Journal of Mathematical Analysis and Applications, 325, 1: 571--585. doi:10.1016/j.jmaa.2006.02.012, preprint

### Project-completion:

This project ended on 01.01.2007. Its completion allowed the sucessfull application for a 'High Potential'-Project of the WWTF, see MULAC.