Handbook of Linear Algebra, 2nd Edition

Chapter 81

Fourier Analysis

Kenneth Howell

University of Alabama in Huntsville

81.1 Introduction

Fourier analysis has been employed with great success in a wide range of applications. The underlying theory is based on a small set of linear transforms on particular linear spaces. It may, in fact, be best to refer to two parallel theories of Fourier analysis—the function/functional theory and the discrete theory—according to whether these linear spaces are infinite or finite dimensional. The function/functional theory involves infinite dimensional spaces of functions on ℝn and can be further divided into two “subtheories”—one concerned with functions that are periodic on ℝn (or can be treated as periodic extensions of functions on finite subregions) ...