- Reproduce the standard definitions used in the theory of Hilbert spaces and bounded linear transformations.
- Compute the Fourier coefficients of vectors with respect to a (possibly infinite) complete, orthonormal basis, in particular for functions on L2[0,T] with a basis formed by harmonic functions.
- Compute the (inverse) Fourier transform of nice enough functions, and interpret in a rudimentary fashion the extension of the Fourier transform from L2(−∞,∞) to bounded functions, using delta functions.
- Reproduce the basic properties of the Fourier series and the Fourier transformations and use these to determine Fourier series and Fourier transforms.
- Interpret filters in terms of convolutions and differential equations, and analyse the effect of filters.
- Reproduce the definitions of the Laplace transformation and the associated properties and use them to compute (inverse) Laplace transforms, and use it to solve linear differential equations.
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In 1807 Jean Baptiste Joseph Fourier, in his desire to determine the heat distribution in a metal plate, came up with the bold idea that every function can be written as a sum of harmonic functions. Understandably his claim was met with scepticism, especially because Fourier did not provide a proof of his claim. Several years later (still within Fourier’s lifetime) Niels Henrik Abel settled the issue with a proper proof: Fourier was right after all, and this sum of harmonic functions since then carries his name: the Fourier series.
Nowadays it is difficult to imagine a world without Fourier series. Trillions of Fourier series are calculated routinely every second. For instance your laptop computes 250000 such series for every second that it is talking wirelessly to your modem, and your modem does the same. Ever wondered what JPEG files are? They are essentially just a collection of Fourier co-efficients that your computer (by computing a Fourier series) can turn into a picture. Fourier is everywhere in audio and video processing (such as MP3, MPEG, et cetera) and the digital revolution would have been way less successful if it were not for the Fourier series and the discovery of the “fast fourier transform” which is an algorithm that can compute Fourier series very efficiently.
Fourier analysis is a standard tool in many engineering sciences. It offers an alternative representation of “signals” and “systems” providing lots of insight, and the above mentioned set of applications are just a few of them. Fourier analysis plays a very important role in signal processing applications, it is the work-horse behind efficient algorithms for computation of products of very long integers, and is used to solve partial differential equations, and much more.
In this course we analyze the Fourier series in detail and we use it to solve a number of “signal processing” problems and — which is how it all started — to solve certain differential equations and partial differential equations. The course also introduces the Laplace transform, which can be seen as an extension of the Fourier transform.
The material presented in Chapters 2–5 is quite common in engineering sciences. The first chapter, however, is of a different nature and might fit in an advanced course on linear algebra or a first course on functional analysis. It introduces Banach and Hilbert spaces and more, providing an abstract and general view of Fourier series.
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