From Linear Algebra to Gabor Analysis: a short course consisting of three units

Gabor Analysis is usually understood as the non-orthogonal expansions of functions, signals or distributions as double sums, using building blocks which are time-frequency shifts of a so-called Gabor atom or building block, often a Gaussian. This setting requires a mix of methods from Fourier analysis and general functional analysis, which sometimes obscures the underlying algebraic structures. Hence we start with Gabor Analysis of finite, discrete signals, i.e. functions on the cyclic group of order $n$, and explain the relevant concepts in the context of Linear Algebra.

Thus the first unit recalls basic concepts from linear algebra, sometimes from a non-standard view-point. Starting from matrix-vector multiplication as a way to produce linear combinations of column vectors we discuss generating systems, linear independence, the SVD (singular value decomposition), Loewdin orthogonalization which will be later connected with the Hilbert space concept of frames and Riesz sequences resp. Riesz bases. Especially the situation of shift-invariant systems, discrete versions of (e.g. cubic) B-splines and the discrete version of a Gauss function can be discussed, with a particular emphasize on the DFT (realized as FFT, the Fast Fourier Transform). The main goal of this first section will be to understand the pseudo-inverse (related to the solution of the minimal norm least square solution of $A * x = b$), specifically in the context of Gabor analysis, where the generating family is obtained by the use of cyclic shifts in the time and in the Fourier domain (or translation operator followed by multiplication with pure frequencies).

In the second part of the short-course we will take the situation to the continuous variable case, i.e. we will discuss Gabor Analysis as well as the foundations of time-frequency analysis for functions on the Euclidean spaces $R^d$. The natural starting point is the Hilbert space $L^2(R^d)$, but soon it turns out that one needs subspaces which allow to control the boundedness of linear operators arising in this context. It turns out that the Segal-algebra $S_0(R^d)$ (Feichtinger’s algebra) is most useful in this context and allows to transfer results which are easy to understand in the finite-dimensional setting to the infinite dimensional setting. Together with its dual space, nowadays called the space of mild distributions, one has the so-called Banach Gelfand Triple $(S_0,L_2,S_0*)$, which consists of all the tempered distributions which have a short-time Fourier transform $V_g(\sigma)$ in $(L_1,L_2,L_infty)$
respectively. These spaces are special cases of the larger family of so-called modulation spaces. Among others it will be explained that the elements of $S_0$ are those with absolutely convergent Gabor expansions, while $S_0*$ is equivalent to the existence of a Gabor series expansion with bounded coefficients (and $L_2$ with square summable coefficients).

The third unit be dedicated to a closer study of the world of mild distributions, sufficient and necessary conditions for a continuous and
integrable function to belong to $S_0$, and so on. It will be shown how the use of the Banach Gelfand Triple allows to transform the structure obtained in the finite dimensional setting to the full generality (in fact, to one could perform Gabor Analysis then over LCA = locally compact Abelian groups. Using appropriate diagrams one can explain in which sense the notation of frames generalized the concept of generating families (potential redundant systems), while the concept of Riesz sequences extends to the idea of linear independent families. The Balian-Low Theorem implies that that – in contrast to the wavelet case – there are no orthonormal bases of Gabor type with ``good atoms’’ (meaning in $S_0$). As time permits also other facts from classical or applied Fourier analysis (such as the Shannon sampling theorem, or the characterization of translation invariant linear systems) will be discussed.