Classical Fourier Analysis, a foundational aspect of Functional Analysis, heavily relies on the properties of function spaces defined by integrability conditions. These function spaces are instrumental in the progression of classical Fourier methods. Abstract Harmonic Analysis unifies results first obtained for periodic and then for square integrable functions on Euclidean spaces. By starting with a locally compact Abelian group, endowed with a Haar measure, one can introduce the corresponding notion of Fourier transforms. In the case of finite Abelian groups, this yields the classical Discrete Fourier Transform (DFT) and its efficient computation through the Fast Fourier Transform (FFT).

From a practical standpoint, such theoretical considerations have had limited direct impact on the development of modern signal processing algorithms, such as the MP3 compression scheme widely used for audio signals. These signals are not truly L^2-functions but instead have a (discrete-time) short-time Fourier transform (STFT), based on the digital version of the signal, sampled at 44,100 samples per second. This scenario necessitates different function spaces.
Collaboration with engineers (e.g. in the area of mobile communication) has reconfirmed a gap in the understanding of the necessary mathematical framework.

Bridging this gap has led to the development of new mathematical methods over the past 40 years, creating a dynamic field of mathematical analysis, known as Time-Frequency Analysis, with Gabor Analysis as a central component. This field utilizes series expansions of non-periodic signals (such as music) into localized Fourier series expansions. The relevant function spaces in this context are modulation spaces, with "Feichtinger's algebra" and its dual, known as the space of "mild distributions," being the most pertinent. Among others one can discuss new and interesting approximation theoretic questions relevant for applications.