Fourier Analysis

This chapter covers the Fourier series, the Fourier transform, and the discrete Fourier transform. A novel feature included here is the reason why the fast Fourier transform works—the underlying group theory is explained behind its performance. It further includes sampling theory, Whittaker-Shannon sampling theorem, and aliasing in detail. For this chapter, a rudimentary knowledge of Fourier transforms at the undergraduate level is assumed, and it gives the modern advanced perspective of the theory. For example, the Fourier transform is viewed as a change of bases in the \(L^2\) space. The reconstruction of a function from its Fourier coefficients—the convergence of the Fourier series—pointwise and in the \(L^2\) norm, Parseval’s identity as a consequence of the unitarity of the Fourier transform operator, etc., are some of the features that are discussed in this chapter.