Herausforderungen der Theorie der Rahmen Multiplikatoren

Theoretical framework:
The multiplier concept is a very natural one that occurs in a lot of scientific questions in signal processing, psychoacoustics, physics, mathematics, and other disciplines. Multipliers have been used for quite some time implicitly in applications, while their deep theoretical investigation in relation to frame theory has become a research focus only in the last two decades. Some work on inversion of multipliers was done in the last decade, but still many questions have remained unanswered and many questions in new directions have arisen. In this project we will perform application-oriented fundamental research on multipliers focused on several directions which are of interest from a theoretical point of view, as well as of relevance and importance for applications.

Research questions / objectives:
The main research questions and goals of the project are: to solve a challenging conjecture concerning an intuitive representation of unconditionally convergent multipliers (UCM); to investigate inversion of multipliers for structured frames; to develop novel efficient algorithms for inversion of multipliers and to implement them in LTFAT; to investigate generalized inversion of multipliers and to apply results from this study to information theory; to investigate eigenvalues and eigenvectors of multipliers through a novel approach in relation to time-frequency representation of signals; to determine multipliers whose invertibility can be lifted to certain classes of Banach/Frechet/distribution spaces.

Approach / methods:
We will use advanced techniques from functional analysis, operator theory, frame theory, numerical analysis, and other related fields, building on the previous work and experience of the project applicant and the cooperation partners, as well as will work on inventing and developing novel approaches.

Level of originality / innovation:
Some particularly innovative points of the project include: novel ideas and original techniques for dealing with the challenging conjecture for UCM; discovery of new sufficient conditions for invertibility of multipliers, new simple formulas for the inverse operator, and new efficient algorithms for inversion; new research questions aiming discovery of useful expressions of the special sequences, related to a representation of the inverse multiplier, and invention of iterative algorithms to approximate these sequences; novel research line for investigation of generalized inversion of frame multipliers and application of the developed formulas and algorithms to communication channels; novel ideas and techniques for relating the topic of eigenvalues and eigenvectors of multipliers to time-frequency representation of signals; research on new challenging questions for lifting of invertibility.

Primary researchers involved:
The PI is Diana Stoeva who has deep expertise and research experience in frame theory and on multipliers.