Chapter 4: Introduction to the Fourier Transform

In this chapter we introduce the Fourier transform and review some of its basic properties. The Fourier transform is the “Swiss army knife” of mathematical analysis; it is a sturdy general-purpose tool with many useful special features. The Fourier transform makes precise the concept of decomposing a function into its harmonic components. In engineering it is used to define the power spectrum and describe many filtering operations. Because it has efficient numerical approximations, it forms the foundation for most image and signal processing algorithms. Using its properties as a linear transformation of infinite-dimensional, normed vector spaces, we define the Fourier transform and its inverse for several different spaces of functions. Among other things we establish the Parseval formula relating the energy content of a function to that of its Fourier transform. We also study the connections between the smoothness and decay of a function and that of its Fourier transform. In marked contrast to the Radon transform, the theory of the Fourier transform is largely independent of the dimension: The theory of the Fourier transform for functions of one variable is formally the same as the theory for functions of 2, 3, or n variables. For simplicity we begin with a discussion of the basic concepts for functions of a single variable, though in some definitions, where there is no additional difficulty, we treat the general case from the outset. The chapter ends with a recapitulation of the main results for functions of n variables. 4.1 The Complex Exponential Function See: 2.3, A.3.1. The building block for the Fourier transform is the complex exponential function, eix.