Explore the foundations of Gabor analysis through a MATLAB-based lecture that delves into Fourier Analysis via the Banach Gelfand Triple. Gain insights into Abstract Harmonic Analysis, focusing on functions over finite Abelian groups and their representation as matrices. Learn how to apply matrix analysis methods for optimal estimates and discover the parallels between discrete and continuous cases. Engage with practical experiments using MATLAB, OCTAVE, or the LTFAT toolbox to develop conjectures and ideas applicable to continuous scenarios. Cover topics including wavelets, convolution, mobile channels, minimal Banach space, classical Fourier analysis, Hilbert space, test functions, category theory, topology, and time-frequency analysis. Enhance your understanding of signal processing mathematics through this comprehensive 59-minute presentation from the Hausdorff Trimester Program.