Harmonic analysis is the branch of mathematics which developed from the study of Fourier series, Fourier transforms, and other related operators. One of the most basic problems in harmonic analysis is to determine how periodic functions can be written as a sum of trigonometric functions. Suppose, for example, that ƒ(x) is a real or complex valued function defined for all real numbers x, and ƒ is also periodic with period 1. By periodic with period 1 we mean that ƒ satisfies the identity ƒ(x+1) = ƒ(x) for all real numbers x. It follows then that ƒ(x+n) = ƒ(x) for all integers n. By considering a graph of y = ƒ(x) it is apparent that ƒ is completely determined by its behavior on any interval of length 1, for example, the interval [0,1]. Now some of the simplest examples of periodic functions with period 1 are the trigonometric...