Executive Summary : | Time-frequency analysis has gained significant attention in recent years, particularly in the field of modulation spaces. This project aims to address a problem about positive semi-definite trace-class operators on a separable Hilbert space posed by Heil and Larson in 2008. The project also investigates irregular Gabor frames on modulation spaces, explores the Balian-Low theorem, and several uncertainty principles on modulation spaces using various transforms. The project also investigates the Feichtinger problem, which focuses on positive semi-definite trace-class pseudodifferential operators on modulation spaces. The project also investigates irregular Gabor frames on modulation spaces, taking window functions from $M^1$ and analyzing whether the $M^1$ condition can be relaxed to the Wiener space $W(L^\infty, L^1)$. The project also investigates problems for infinite-dimensional spaces, such as classifying invertible operators for invariant frame operators, mapping a frame to an equal norm frame, and making an equal norm Parseval frame. The project also investigates the Beurling, Donoho--Stark, Gelfand--Shilov, Miyachi, and Slepian--Pollak theorems on modulation spaces using various transforms, including Opdam--Cherednik transforms, generalized Fourier transforms, Dunkl transforms, and Hankel transforms. In conclusion, this project aims to explore the various aspects of time-frequency analysis, including modulation spaces, and explore the Balian-Low theorem and uncertainty principles using various transforms. |