Within Classical Fourier analysis (Fourier inversion, Gabor expansions, conditional convergence, etc.) the fact that one is dealing with infinite dimensional spaces, equivalence classes measurable functions or generalized functions requires sophisticated concepts of functional analysis. Using norms, function spaces, distributions and operators, Hilbert spaces one can control the situation.

Most of these questions appear to evaporate as soon as one switches to finite dimensional vector spaces, i.e. to the setting of complex-valued vectors of finite length and the realm of MATLAB (or similar mathematical software packages), because sums are finite, Cauchy sequence ``stagnate'' as soon as numerical precision is reached, etc.. On the other hand question concerning precision, condition numbers, computational costs or memory requirements start to play a role.

We will propose to take a tour d'horizon through the landscape of abstract harmonic analysis, where the setting of LCA (locally compact Abelian) groups provides a joint language to pure frequencies, plane waves and their generalization. But instead of studying the analogy between the different setting (there is always
a dual group, a natural form of convolution and a Fourier transform diagonalizing this commutative multiplication), we will investigate the connections between them. As a prototypical example let us remind of the fact that it is by far not obvious how to relate (except just heuristically) the Fourier transform on the real
line to the FFT (the fast Fourier transform), which is well-known for its numerical efficiency.

Thus the main theme will be the approximation of the continuous reality (objects of interest over Euclidean spaces $\Rst^d$, for example, [pseudo-differential] operators, signal expansions, etc.)
by means of finite-dimensional, constructive and computable (realizable) computations. This requires concepts from functional analysis (here the recent concept of Banach Gelfand Triples based on the Segal algebra $\SORd$ is of great use) on the one hand, but on the other hand also efficient algorithms which are not limited to signals of modest length.

The course is supposed to provide both theoretical insight into the situation and practical knowledge about the implementation of such approximations using the MATLAB toolboxes developed at NuHAG.