Harmonic Analysis is concerned with the decomposition of functions, signals respectively distributions over locally compact Abelian groups, such as Rd, or in the classical case over the torus (Fourier series expansions of periodic functions) or for modern applications over cyclic groups of order N respectively their products (for example, for image processing applications), using discrete Fourier transform methods. The building blocks are the pure frequencies (or plane waves, etc.) which are eigenvectors for the translation operator. Because translation invariant operators arise very naturally in many applied situations (mechanical system, transmission systems), their all important description as convolution operators and the fact that the corresponding Fourier transform allows to turn convolution into pointwise multiplication are at the core of Fourier Analysis, which is by now a
very mature and sophisticated field. In the last 30 years a mathematical theory, called time-frequency analysis arose, which is based on the idea of a localized Fourier transform using typically a finite length window. So instead of having an infinitely fine frequency domain with no time information Time Frequency analysis provides a picture comparable to score)
showing how the (!discrete) harmonic decomposition of a signal changes over time. This theory, often called Gabor Analysis requires new tools and gives rise to new and interesting mathematical questions. It turned out, that a specific Banach space of continuous and integrable test functions (which can be defined over any locally compact Abelian group, specifically on Rd) together with its dual space provide an appropriate framework for the description of both the classical situation as well as the new setting. In the course the participants will be lead from linear algebra, with some functional analysis to these modern foundations of a classical field.