Hermite Multiplier, Convolutions and Time-Frequency Analysis

The concept of convolution is very important in both mathematical Fourier Analysis and engineering, such as in analyzing translation-invariant linear systems. It helps us understand how signals change in different systems, like electronic circuits or signal processing algorithms. Convolution is closely related to the Fourier transform in its various forms. This project aims to study different versions of convolution that connect Hermite functions with Time-Frequency Analysis (TFA). By doing this, it will solidify and extend the findings from engineering papers and existing literature. This project is innovative because it combines questions and methods from different scientific fields. Mathematically, it explores Gabor and Hermite expansions of tempered distributions, which include various function spaces and multipliers. These concepts are well-established in this context. A key part of this project is studying the fractional Fourier transform (FrFT), which transforms a signal from its original domain (time or frequency) to a new domain defined by a fractional angle. This tool is valuable for analyzing signals in ways that traditional Fourier or time- domain methods cannot. Notable applications of FrFT include optics such as optical signal processing, phase retrieval, etc. We plan to integrate mathematical theory with real-world applications to enhance our understanding and practical skills in engineering. This will advance the method of Conceptual Harmonic Analysis which aims at the developments of algorithms and numerical tools for applications but also supports theoretical concepts through numerical exploration. This project will also explore different types of multipliers, like Gabor multipliers or Hermite-(pseudo)- multipliers, and their use in specific Banach spaces of distributions, known as modulation spaces (introduced by the host) or Hermite-Besov spaces. We will investigate their mutual approximations and their effects on function spaces defined by other expansion types and various discrete versions.