Time-frequency analysis

Abstract

This thesis presents a detailed study of time-frequency analysis, a modern area within harmonic analysis concerned with the simultaneous treatment of time and frequency information. The work begins with foundational concepts from measure theory and functional analysis, including Lp spaces and bounded linear operators. We then explore essential tools from Fourier analysis, setting the stage for a deeper investigation of time-frequency methods. The uncertainty principle is introduced to highlight the inherent limitations of simultaneous time and frequency localization. Building on this, the short-time Fourier transform is developed as a central tool, alongside other time-frequency representations such as the Wigner distribution, spectrogram, and the ambiguity function. We then delve into Gabor analysis and frame theory, emphasizing the construction and existence of Gabor frames in L2(Rd), supported by results such as Walnut’s representation. Throughout, the thesis focuses on the mathematical structures underlying time-frequency analysis rather than its applications, aiming to provide a rigorous and self-contained exposition of the subject.