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Unformatted text preview: Chapter 8 Fourier Transforms Fourier series and their ilk are designed to solve boundary value problems on bounded intervals. The extension of Fourier methods to the entire real line leads naturally to the Fourier transform , an extremely powerful mathematical tool for the analysis of non- periodic functions. The Fourier transform is of fundamental importance in a broad range of applications, including both ordinary and partial differential equations, quantum me- chanics, signal and image processing, control theory, and probability, to name but a few. In this chapter, we begin by motivating the construction by investigating how Fourier series behave as the length of the interval goes to infinity. The resulting Fourier transform maps a function defined on physical space to a function defined on the space of frequencies, whose values quantify the amount of each periodic frequency contained in the original function. The inverse Fourier transform then reconstructs the original function from its transform. The integrals defining the Fourier transform and its inverse are remarkably alike, and this symmetry is often exploited, for example when assembling tables of Fourier transforms. One of the most important properties of the Fourier transform is that it converts calculus: differentiation and integration into algebra: multiplication and division. This underlies its application to linear ordinary differential equations and, in the following chapters, partial differential equations. In engineering applications, the Fourier transform is sometimes overshadowed by the Laplace transform, which is a particular subcase. The Fourier transform is used to analyze boundary value problems on the entire line. The Laplace transform is better suited to solving initial value problems, [ 23 , 40 ], but will not be developed in this text. The Fourier transform is, like Fourier series, completely compatible with the calculus of generalized functions. The final section contains a brief introduction to the analytical foundations of the subject, including the basics of Hilbert space. However, a full, rigorous development requires more powerful analytical tools, and the interested reader is therefore referred to more advanced texts, e.g., [ 44 , 77 , 137 ]. 8.1. The Fourier Transform. We begin by motivating the Fourier transform as a limiting case of Fourier series. Although the rigorous details are rather exacting, the underlying idea can be straightfor- wardly explained. Let f ( x ) be a reasonably nice function defined for all < x < . The goal is to construct a Fourier expansion for f ( x ) in terms of basic trigonometric func- tions. One evident approach is to construct its Fourier series on progressively longer and 1/19/12 281 c circlecopyrt 2012 Peter J. Olver longer intervals, and then take the limit as their lengths go to infinity. This limiting process converts the Fourier sums into integrals, and the resulting representation of a function is renamed the Fourier transform. Since we are dealing with an infinite interval, there are norenamed the Fourier transform....
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