A Guided Tour from Linear Algebra to the Foundations of Gabor Analysis

Starting from the context of finite dimensional signal spaces, as it can be constructively realized on a computer and completely described in terms of concepts from linear algebra, we describe first the algebraic side of this problem. The second part of the material emphasizes the necessary definitions and problems (e.g., the form of convergence of various infinite series occurring during the discussion) and provides a description of tools that have become relevant within modern time-frequency analysis in order to overcome the technical problems arising in the context of Gabor analysis involving a continuous domain (such as the Euclidean space). As it is shown, the so-called Banach Gelfand Triple built upon the Segal Algebra S0(R) appears as the appropriate and universally useful tool for a clear treatment of questions of time-frequency analysis, even if the interest is mostly in operators over L. On the other hand a family of function (or distribution) spaces defined by means of timefrequency tools (the so-called modulation spaces) appear to provide a suitable frame-work for a more refined discussion of mathematical problems arising in this field.