ABSTRACT
This chapter outlines some ideas of material that should be covered by an application-oriented course on Fourier Analysis. It also outlines the relevant building blocks for a possible course preparing the ground for a better understanding of the mathematical foundations of signal processing, with an emphasis on the functional analytic side. An important part of modern digital signal processing is based on the use of computers. It is a plain statement that a computer—at least if viewed as a number-crunching machine—can only digest a finite set of vectors of finite length. One of the central topics of Fourier Analysis is clearly the concept of “convolution” and the so-called “convolution theorem,” essentially stating that the Fourier transform is turning convolution into pointwise multiplication (and vice versa) which are used to characterize translation-invariant linear systems as a kind of moving average or convolution.
