Abstract
The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating images in computer tomography. In order to achieve this, appropriate algorithms are necessary. In this context, the fast Fourier transform (FFT) plays a key role which is an algorithm for calculating the discrete Fourier transform (DFT); this, in turn, is tightly connected with the discrete Fourier series. The last one itself is the discrete analog of the common (continuous-time) Fourier series and is usually learned by mathematics students from a theoretical point of view. The aim of this expository/pedagogical paper is to give an introduction to the discrete Fourier series for both mathematics and engineering students. It is intended to expand the purely mathemati...
Highlights
Introduction to the discreteFourier series considering both mathematical and engineering aspects - A linear algebra approachLudwig Kohaupt1,i* AbstractThe discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike
The subject-specific aim of the present paper is to introduce the discrete Fourier series taking into account both mathematical and engineering aspects
As already mentioned, it wants to give an introduction to the discrete Fourier series in a way that emphasizes a unifying approach to problem solving by beginning with a more general problem in a vector space endowed with a scalar product
Summary
There is a strong interaction between the development of the mathematical sciences, on the one hand, and the application of mathematical tools in other sciences such as in physics, medicine, or engineering, on the other hand. As already mentioned, it wants to give an introduction to the discrete Fourier series in a way that emphasizes a unifying approach to problem solving by beginning with a more general problem in a vector space endowed with a scalar product It is possible and sometimes even advisable to study the approximation of periodic functions by trigonometric polynomials on an interval or a discrete point set as well as the best approximation of sufficiently smooth functions first, and afterwards investigate the more general case of the best approximation of a vector in a finite-dimensional vector space by a linear combination of linearly independent vectors. The complex form often is mathematically simpler than the real form
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