Challenges in Frame Multiplier Theory

The concept of a multiplier is a natural one that arises from applications and occurs in many scientific questions in various disciplines like signal processing, mathematics, physics, and other. In signal processing, roughly speaking one can think about a multiplier as a tool for a signal modification and it can be described in three steps. First, the signal is visualized via some transformation so that certain properties of the signal (e.g. the frequencies over time, the energy of the signal) can be easily found or seen. Then one can deal with the visualization of the signal to modify some features of interest. Finally, the modified image is transformed back to the signal domain and thus a new, modified signal is obtained. In practice, what a sound engineer does during a concert, operating an equalizer, is actually a manipulation of frequency ranges in real time in order to improve the sound quality. Multipliers can also apply to separation tasks, e.g. when aiming to separate the singer`s voice from the musical instruments in a song, or when aiming to extract the sound of some animal from a noisy record in the nature, etc. When applying a multiplier to get a new modified signal, it is not always possible to convert back to the original signal. It is of relevance and importance to determine multipliers whose action can be inverted, as well as to be able to describe the inverse operation (the one that transforms the modified signal back to the original one) and to compute the inversion efficiently. While multipliers have long been used implicitly in applications, their in-depth theoretical study in relation to frame theory became a research focus only in the last two decades. Some work on inversion of frame 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 an in-depth study on challenging questions on multipliers that concern inversion and related topics. Some of the main aims are to investigate invertibility of specific classes of multipliers, relevant for applications, to determine new simple formulas for the inverse operation, and to develop and implement novel efficient algorithms for the inversion; to solve a 12-years old conjecture about a representation of an important class of multipliers; to develop novel approaches for investigation of various other characteristics of the multiplier operator, in particular, on eigenvalues and eigenvectors.