Chapter 13 Fourier Analysis

In addition to their inestimable importance in mathematics and its applications, Fourier series also serve as the entry point into the wonderful world of Fourier analysis and its wide-ranging extensions and generalizations. An entire industry is devoted to developing the theory and enlarging the scope of applications of Fourier–inspired methods. New directions in Fourier analysis continue to be discovered and exploited in a broad range of physical, mathematical, engineering, chemical, biological, financial, and other systems. In this chapter, we will concentrate on four of the most important variants: discrete Fourier sums leading to the fast Fourier transform; the modern theory of wavelets; the Fourier transform; and, finally, its close relative, the Laplace transform. In addition, more general types of eigenfunction expansions associated with partial differential equations in higher dimensions will appear in the following chapters. Modern digital media, such as CD’s, DVD’s and MP3’s, are based on discrete data, not continuous functions. This requires the development of a purely discrete version of Fourier series, in which one replaces periodic functions, representing analog signals, by their sample vectors. The resulting discrete Fourier representation can, in fact, be handled by finitedimensional vector space methods, and so, technically, belongs back in the linear algebra portion of this text. However, the insight gained from the classical continuous Fourier theory proves to be essential in understanding and analyzing its discrete digital cousin. An important application of discrete Fourier sums is in signal and image processing. One typically samples a signal at equally spaced time intervals, and then works exclusively with the resulting discrete (digital) data. Basic data compression and noise removal algorithms are applied to the sample’s discrete Fourier coefficients, acting on the observation that noise tends to accumulate in the high frequency Fourier modes, while most important features are concentrated at low frequencies. In Section 13.1, we develop the basic Fourier theory in this discrete setting, culminating in the Fast Fourier Transform, which produces an efficient numerical algorithm for passing between a signal and its discrete Fourier coefficients, which has become an essential tool in modern signal processing and numerical analysis. One of the inherent limitations of classical Fourier methods, both continuous and discrete, is that they are not well adapted to localized data, stemming from the inherent lack of localization in the constituent trigonometric functions. (In physics, this lack of localization is the basis of the Heisenberg Uncertainty Principle.) As a result, Fourier-based signal processing algorithms can become inaccurate and/or inefficent when confronting highly localized signals or images. In the second section, we introduce the modern theory of wavelets, which is a recent extension of Fourier analysis that more naturally incorporates multiple scales and localization. Wavelets are playing an increasingly dominant role in